29
Hindawi Publishing Corporation ISRN Mathematical Physics Volume 2013, Article ID 149169, 28 pages http://dx.doi.org/10.1155/2013/149169 Review Article Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences T. D. Frank Center for the Ecological Study of Perception and Action, Department of Psychology, University of Connecticut, 406 Babbidge Road, Storrs, CT 06269, USA Correspondence should be addressed to T. D. Frank; [email protected] Received 14 January 2013; Accepted 13 February 2013 Academic Editors: G. Cleaver, G. Goldin, and D. Singleton Copyright © 2013 T. D. Frank. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Strongly nonlinear stochastic processes can be found in many applications in physics and the life sciences. In particular, in physics, strongly nonlinear stochastic processes play an important role in understanding nonlinear Markov diffusion processes and have frequently been used to describe order-disorder phase transitions of equilibrium and nonequilibrium systems. However, diffusion processes represent only one class of strongly nonlinear stochastic processes out of four fundamental classes of time-discrete and time-continuous processes evolving on discrete and continuous state spaces. Moreover, strongly nonlinear stochastic processes appear both as Markov and non-Markovian processes. In this paper the full spectrum of strongly nonlinear stochastic processes is presented. Not only are processes presented that are defined by nonlinear diffusion and nonlinear Fokker-Planck equations but also processes are discussed that are defined by nonlinear Markov chains, nonlinear master equations, and strongly nonlinear stochastic iterative maps. Markovian as well as non-Markovian processes are considered. Applications range from classical fields of physics such as astrophysics, accelerator physics, order-disorder phase transitions of liquids, material physics of porous media, quantum mechanical descriptions, and synchronization phenomena in equilibrium and nonequilibrium systems to problems in mathematics, engineering sciences, biology, psychology, social sciences, finance, and economics. 1. Introduction 1.1. McKean and a New Class of Stochastic Processes. In two seminal papers, McKean introduced a new type of stochastic processes [1, 2]. e stochastic processes studied by McKean satisfy driſt-diffusion equations for probability densities that are nonlinear with respect to their probability densities. While nonlinear driſt-diffusion equations have frequently been used to describe how concentration fields evolve in time [3], the models considered by McKean describe stochastic processes. A stochastic process not only describes how expectation values (such as the mean value) evolve in time, but also, provides a complete description of all possible correlations between two or more than two time points [46]. Consequently, the study of stochastic processes goes beyond the study of concentration fields or the evolution of single-time point expectation values such as the mean. e observation by McKean that nonlinear driſt-diffusion equations can describe stochastic processes opened a new avenue of research. 1.2. From McKean’s Stochastic Processes to Strongly Nonlinear Stochastic Processes. Due to the key role that the nonlinearity has for the processes considered by McKean and due to the fact that the processes satisfy the Markov property, processes of this type have been called “nonlinear Markov processes” (see, e.g., [7]). Moreover, in this context the term “nonlinear Fokker-Planck equations” has also been introduced because the driſt-diffusion equations of nonlinear Markov processes exhibit formally the mathematical structure of Fokker-Planck equations. However, the phrases “nonlinear Markov pro- cesses” and “nonlinear Fokker-Planck equations” are mis- leading because they refer both to processes that are subjected to nonlinear forces and to processes that are described by driſt-diffusion equations that are nonlinear with respect to their probability densities. In search for a more specific label,

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Hindawi Publishing CorporationISRNMathematical PhysicsVolume 2013 Article ID 149169 28 pageshttpdxdoiorg1011552013149169

Review ArticleStrongly Nonlinear Stochastic Processes inPhysics and the Life Sciences

T D Frank

Center for the Ecological Study of Perception and Action Department of Psychology University of Connecticut406 Babbidge Road Storrs CT 06269 USA

Correspondence should be addressed to T D Frank tillfrankuconnedu

Received 14 January 2013 Accepted 13 February 2013

Academic Editors G Cleaver G Goldin and D Singleton

Copyright copy 2013 T D Frank This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Strongly nonlinear stochastic processes can be found in many applications in physics and the life sciences In particular in physicsstrongly nonlinear stochastic processes play an important role in understanding nonlinear Markov diffusion processes and havefrequently been used to describe order-disorder phase transitions of equilibrium and nonequilibrium systems However diffusionprocesses represent only one class of strongly nonlinear stochastic processes out of four fundamental classes of time-discrete andtime-continuous processes evolving on discrete and continuous state spaces Moreover strongly nonlinear stochastic processesappear both as Markov and non-Markovian processes In this paper the full spectrum of strongly nonlinear stochastic processes ispresented Not only are processes presented that are defined by nonlinear diffusion and nonlinear Fokker-Planck equations but alsoprocesses are discussed that are defined by nonlinearMarkov chains nonlinear master equations and strongly nonlinear stochasticiterative maps Markovian as well as non-Markovian processes are considered Applications range from classical fields of physicssuch as astrophysics accelerator physics order-disorder phase transitions of liquids material physics of porous media quantummechanical descriptions and synchronization phenomena in equilibriumandnonequilibrium systems to problems inmathematicsengineering sciences biology psychology social sciences finance and economics

1 Introduction

11 McKean and a New Class of Stochastic Processes Intwo seminal papers McKean introduced a new type ofstochastic processes [1 2] The stochastic processes studiedby McKean satisfy drift-diffusion equations for probabilitydensities that are nonlinear with respect to their probabilitydensities While nonlinear drift-diffusion equations havefrequently been used to describe how concentration fieldsevolve in time [3] themodels considered byMcKean describestochastic processes A stochastic process not only describeshow expectation values (such as the mean value) evolve intime but also provides a complete description of all possiblecorrelations between two or more than two time points[4ndash6] Consequently the study of stochastic processes goesbeyond the study of concentration fields or the evolutionof single-time point expectation values such as the meanThe observation by McKean that nonlinear drift-diffusion

equations can describe stochastic processes opened a newavenue of research

12 From McKeanrsquos Stochastic Processes to Strongly NonlinearStochastic Processes Due to the key role that the nonlinearityhas for the processes considered by McKean and due to thefact that the processes satisfy the Markov property processesof this type have been called ldquononlinear Markov processesrdquo(see eg [7]) Moreover in this context the term ldquononlinearFokker-Planck equationsrdquo has also been introduced becausethe drift-diffusion equations of nonlinear Markov processesexhibit formally themathematical structure of Fokker-Planckequations However the phrases ldquononlinear Markov pro-cessesrdquo and ldquononlinear Fokker-Planck equationsrdquo are mis-leading because they refer both to processes that are subjectedto nonlinear forces and to processes that are described bydrift-diffusion equations that are nonlinear with respect totheir probability densities In search for a more specific label

2 ISRNMathematical Physics

two alternative terms have been proposed in the literatureWehner and Wolfer [8] suggested the term ldquotruly nonlinearFokker-Planck equationsrdquo whereas Frank [9] coined the termldquostrongly nonlinear Fokker-Planck equationsrdquo Only recentlyit has become clear that the class of stochastic processesaddressed by McKean is not limited to be described by drift-diffusion equations of the Fokker-Planck type Thereforethe research field inspired by the work of McKean is aboutnonlinear stochastic processes in general To give processes ofthis kind amore specific name wemay adopt the suggestionsby Wehner and Wolfer or Frank and say that there is a novelclass of processes called ldquotruly nonlinear stochastic processesrdquoor ldquostrongly nonlinear stochastic processesrdquo In what followswe will use the term ldquostronglyrdquo that should be considered asa synonym to ldquotrulyrdquo just to avoid repeating the phrase ldquotrulyor stronglyrdquo over and over again

For the purpose of this paper the following unifying def-inition of a strongly nonlinear process will be used

A strongly nonlinear stochastic process is a sto-chastic process for which the evolution of realiza-tions depends on at least one expectation valueof the process or on the appropriately definedprobability density of that process

Note that for strongly nonlinear processes evolving on dis-crete state spaces the realizations are required to depend on atleast one expectation value or on the occupation probabilityrather than the probability density However as we willdemonstrate in Section 2 for our purposes the occupationprobability of a discrete-state process can be derived froman appropriately defined probability density Therefore thedefinition as given above holds both for processes evolvingon continuous and discrete state spaces

In this review of strongly nonlinear stochastic processesa distinction will be made between four classes of processesThese classes arise naturally if it is taken into considerationthat in many disciplines time is not necessarily considered tobe a continuous variable Rather time might be consideredas a discrete variable such that processes evolve in discretetime steps For example a process describing the evolutionof a stock price on a day-to-day basis is typically modelledby means of a time-discrete process [10 11] In short timecan be a discrete or continuous variable Likewise the statespace of a process might be discrete or continuous The fourpossible combinations of time and space characteristics giverise to four classes of stochastic processes We will addressthese classes in separate sections see Table 1

Before we proceed in Sections 3 4 5 and 6 with thesystematic presentation outlined in Table 1 we will presentin Section 2 benchmark examples of processes for each of thefour classes For the examples discussed in Section 2 a unify-ing mathematical definition irrespective of class membershipwill be presented in the final subsection of Section 2 Meanfield theory and themathematical literature will be addressedin Sections 7 and 8 As it turns out all examples reviewedin Sections 2 to 6 exhibit the Markov property ThereforeSection 9 is devoted to the review of somemore recent studieson non-Markovian strongly nonlinear stochastic processes

Table 1 Four classes of strongly nonlinear stochastic processes andsections in which they will be addressed

Time discrete Time continuousSpace discrete Section 3 Section 5Space continuous Section 4 Section 6

In Section 10 some key issues of strongly nonlinear stochasticprocesses will be highlighted

2 Benchmark Examples of Strongly NonlinearProcesses and on a General StochasticEvolution Equation for a Broad Class ofStrongly Nonlinear Markov Processes

21 Some Notations At the present stage it is useful tointroduce some notations in order to give some benchmarkexamples for each of the four classes indicated in Table 1

Time-discrete processes evolve on discrete time steps119905 = 1 2 3 In contrast time-continuous processes aredescribed by a continuous time variable 119905 defined on someinterval For our purposes this interval will go from zero toinfinity meaning that we will consider initial value problemsstarting at an initial time 119905 = 0 and extending in timeinfinitely Note that the symbol 119905 will be used to denote bothdiscrete-valued and continuous time variables

Processes evolving in discrete finite state spaces exhibit anumber of 119873 states labeled by 119896 = 1 2 119873 Let 119883 denotethe state variable of a state discrete process which impliesthat 119883 assumes an integer between 1 and 119873 Moreover let119883119903 denote the 119903th realization of the process A fundamental

(although not complete) stochastic description of processesevolving on finite discrete state spaces is given by theoccupation probabilities 119901(119896) The occupation probability119901(119896) defines the probability that the state 119896 is occupied It canbe computed from a number of 119877 realizations in the limitingcase of 119877 rarr infin like

119901 (119896) = lim119877rarrinfin

1

119877120575 (119883119903

119896) (1)

The function120575(sdot) is theKronecker functionwhich equals 1 for119883119903

= 119896 and equals 0 otherwiseThe probabilities119901(119896) definedby (1) satisfy the normalization condition

119873

sum

119896=1

119901 (119896) = 1 (2)

Processes evolving on finite dimensional continuousstate spaces can be described by a state vector 119883 withcoefficients 119883

1 119883

119873 The elements or coordinates 119883

119896are

defined on certain domains of definition Ω119896 For example

a domain of definition may correspond to the real lineAn important measure characterizing stochastic processesevolving on continuous state spaces is the probability density119875 defined as the ensemble average over the Dirac deltafunction 120575(sdot) Note that the same symbol 120575(sdot) will be used forthe Kronecker function and the Dirac delta function in order

ISRNMathematical Physics 3

to stress the analogies between the classes listed in Table 1Mathematically speaking the probability density 119875 can bedefined by

119875 (119909) = lim119877rarrinfin

1

119877120575 (119883119903

1minus 1199091) sdot sdot sdot 120575 (119883

119903

119873minus 119909119873) (3)

In (3) the variable 119909 is the state vector with coordinates1199091 119909

119873 whereas the elements119883119903

1 119883

119903

119873are the elements

of the 119903th vector-valued realization 119883119903 of the process The

probability density satisfies the normalization condition

int

Ωtot

119875 (119909)

119873

prod

119896=1

119889119909119896= 1 Ωtot =

119873

prod

119896=1

Ω119896 (4)

where Ω119896are the domains of definitions mentioned previ-

ouslyLet us return to processes defined on a discrete state space

In addition we put119873 = 1 and assume thatΩtot = Ω1denotes

the real half-line from 0 to infinity Formally the probabilitydensity 119875(119909) assumes the form

119875 (119909) =

119873

sum

119896=1

119901 (119896) 120575 (119909 minus 119896) (5)

where 119901(119896) are the occupation probabilities of the states 119896defined in (1) In turn the occupation probabilities can becalculated from 119875(119909) like

119901 (119896) = int

119896+120576

119896minus120576

119875 (119909) 119889119909 (6)

where 120576 is a positive number smaller than 1 (eg 120576 = 01)In view of (6) the occupation probabilities 119901(119896)may be con-sidered as functions of an appropriately defined probabilitydensity119875 as stated in Section 1With these notations at handsome benchmark examples of strongly nonlinear stochasticprocesses can be given

22 Strongly Nonlinear Markov Processes Described by Non-linear Markov Chains In general the occupation proba-bilities 119901(119896) of time-discrete stochastic processes evolvingon discrete state spaces constitute a time-dependent vector119901(119905) with coefficients 119901(1 119905) 119901(119873 119905) For the importantspecial case of nonlinear Markov processes described bynonlinearMarkov chains the occupation probabilities satisfy[12]

119901 (119896 119905 + 1) =

119873

sum

119894=1

119879 (119894 997888rarr 119896 119905 119901 (119905)) 119901 (119894 119905) (7)

Here 119879 describes the transition probability matrix of theprocess The matrix is an119873 times119873matrix with coefficients119879(119894 rarr 119896) The coefficients 119879(119894 rarr 119896) describe theconditional probabilities that transitions from state 119894 to state119896 occur given that the process is in the state 119894The conditionalprobabilities 119879(119894 rarr 119896)may depend explicitly on time 119905 suchthat 119879(119894 rarr 119896 119905) A nonlinear Markov process is character-ized by the property that at least one conditional probability

depends at least on one expectation value computed from theprocess at time step 119905 or on one occupation probability119901(119896 119905)In this case we have 119879(119894 rarr 119896 119905 119901(119905)) as indicated in (7) Acomplete description of the stochastic process can be derivedfrom (7) as shown in Frank [12] In view of the definition ofa strongly nonlinear stochastic process given in Section 12that is based on realizations of stochastic processes it is worthwhile to consider an alternative definition of the process Tothis end the stochastic map 119891(sdot 120576) can be defined that mapsa state119883(119905) to a state119883(119905 + 1) like

119883 (119905 + 1) = 119891 (119883 (119905) 119905 119901 (119905) 120576 (119905)) (8)

The mapping 119891(sdot 120576) depends on a random variable 120576(119905) thatis uniformly distributed on the interval [0 1] at any timestep 119905 Across time 120576(119905) is statistically independent Thestochastic map 119891(sdot 120576) describes the jumps from119883(119905) to119883(119905+1) satisfying the transition probabilities 119879(119894 rarr 119896) In orderto derive an explicit definition of 119891(sdot 120576) we first note thatthe conditional probabilities 119879(119894 rarr 119896) are by definition realnumbers between 0 and 1 that add up to unity Next it ususeful to consider the cumulative conditional probabilities119878(119894 119896) defined for 119894 = 1 2 119873 and 119896 = 0 1 119873 by

119878 (119894 119896) =

119896

sum

119895=1

119879 (119894 997888rarr 119895 119905 119901 (119905)) (9)

for 119896 gt 0 For 119896 = 0 we put 119878(119894 0) = 0 For 119896 gt 0

the cumulativemeasures 119878(119894 119896) describe the probabilities thattransitions from state 119894 to one of the states 1 2 119896 occur attime 119905 given that the process has an occupation probabilityvector 119901 at time 119905 The cumulative measures represent aset of monotonically increasing points on the interval [0 1]such that the interval between two adjacent points 119878(119894 119896 minus

1) and 119878(119894 119896) equals 119879(119894 rarr 119896) Consequently the pointspectrum 119878(119894 119896) in combinationwith the randomvariable 120576(119905)can be used to construct realizations of the process underconsideration If 119883(119905) = 119894 holds and if 120576(119905) falls into theinterval [119878(119894 119896 minus 1) 119878(119894 119896)) then we put119883(119905+ 1) = 119896 If we doso then transitions from 119894 to 119896 happen with the probability119879(119894 rarr 119896) if the process is in state 119894 at 119905 and distributed like119901(119905) at 119905mdashas required A concise mathematical formulationof this idea can be obtained with the help of the indicatorfunction 119868(119894 rarr 119896 120576) defined by

119868 (119894 997888rarr 119896 119905 119901 120576) = 1 for 120576 isin [119878 (119894 119896 minus 1) 119878 (119894 119896))

0 otherwise(10)

Then the map 119891(sdot 120576) can be defined as follows

119891 (119883 (119905) = 119894 119905 119901 (119905) 120576 (119905)) =

119873

sum

119896=1

119896 sdot 119868 (119894 997888rarr 119896 119905 119901 (119905) 120576 (119905))

(11)

As indicated in (11) the mapping depends on the probabilityvector 119901(119905) because the indicator function 119868(sdot) depends on119901(119905) via the conditional probabilities 119878 and 119879 Thereforeeach realization 119883

119903

(119905) of the process defined by (8)ndash(11)depends on the occupation probabilities119901(119896 119905) of the process

4 ISRNMathematical Physics

The occupation probabilities in turn can be computed fromthe realizations see (1) Therefore the mathematical modelgiven by (1) and (8)ndash(11) provides a closed (and complete)description for the strongly nonlinear stochastic processdescribed by the nonlinear Markov chain (7)

23 Strongly Nonlinear Markov Processes Described by Sto-chastic Iterative Maps The realization of a time-discrete sto-chastic process on an119873-dimensional continuous state spaceis described by a time-dependent state vector119883(119905)with time-dependent components119883

1(119905) 119883

119873(119905) From (3) it follows

that at any time step 119905 the vector119883(119905) is distributed accordingto the time-dependent probability density

119875 (119909 119905) = lim119877rarrinfin

1

119877120575 (119883119903

1(119905) minus 119909

1) sdot sdot sdot 120575 (119883

119903

119873(119905) minus 119909

119873)

(12)

Strongly nonlinear Markov processes defined by stochasticiterative maps with nonparametric white noise satisfy equa-tions of the general form

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(13)

A simplified version of (13) was introduced earlier in Frank[13] In (13) the vector-valued variable ℎ(sdot) is called the driftfunction and describes the deterministic evolution of theprocess Second the term 119892(sdot) sdot 120576(119905) in (13) describes thevector-valued fluctuating force of the stochastic process Therandom vector 120576(119905) with coefficients 120576

1(119905) 120576

119873(119905) is an 119873-

dimensional normally distributed white noise process Thatis each coefficient 120576

119896(119905) is normally distributed at any time

step 119905 and statistically independent in time The randomcoefficients 120576

119896(119905) are statistically independent among each

otherThe variable 119892 is an119873 times119873matrix with coefficients119892119895119896 The coefficients 119892

119895119896with 119896 = 1 119873 for a fixed index

119895 weight the contributions that the random components1205761(119905) 120576

119873(119905) have on the evolution of the process in the

direction (or along the coordinate) 119909119895 In particular for

119892119895119896(sdot 119905) = 0 the random coefficient 120576

119896(119905) does not occur

in the evolution equation of 119883119895at time 119905 Therefore 119892 may

be considered as weight matrix Importantly the coefficients119892119895119896

measure the strength of the fluctuating force 119892(sdot) sdot 120576(119905)of the stochastic process described by (13) The reason forthis is that the random vector 120576(119905) is normalized at any timestep 119905 Most importantly in the case of a strongly nonlinearMarkov process the drift function ℎ(sdot) or the fluctuating force119892(sdot) sdot 120576(119905) or both depend on at least one expectation value oron the probability density 119875 as indicated in (13)The operator120579 maps the probability density 119875 to a real number or a set ofreal numbers This mapping may depend on the value of 119883For example the probability density may be evaluated at thevalue of the process like

120579119883(119905)

[119875 (119909 119905)] = 119875 (119909 119905)|119909=119883(119905)

(14)

In this context it should be noted that the formal depen-dencies ℎ(120579

119883[119875]) and 119892(120579

119883[119875]) include the dependencies on

expectation values (such as the mean) as special cases Forexample

120579119883(119905)

[119875 (119909 119905)] = int

Ωtot

119909119875 (119909 119905)

119873

prod

119896=1

119889119909119896= ⟨119883 (119905)⟩ (15)

where ⟨sdot⟩ is the operator symbol for an ensemble average suchthat ⟨119883⟩ denotes the mean of the random vector 119883 Notethat (12) and (13) provide a closed description of a stochasticprocess Moreover from the structure of (13) it followsimmediately that processes defined by (12) and (13) satisfythe definition of strongly nonlinear stochastic processes pre-sented in Section 12 An illustrative example of a stochasticiterative map (13) is the autoregressive model of order 1subjected to a so-called mean field force ℎMF proportional tothe mean ⟨119883(119905)⟩ of the process For 119873 = 1 the model reads[13]

119883 (119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(16)

with 119892 ge 0 where the parameters119860 and 119861 are scalarsWe willreturn to model (16) in Section 4

24 Strongly Nonlinear Markov Processes Described by Non-linearMaster Equations For time-continuous stochastic pro-cesses evolving on discrete state spaces the occupation prob-abilities 119901(119896) are functions of time 119905 where 119905 is a continuousvariable Likewise the probability vector 119901(119905) becomes atime-dependent variable with coefficients 119901(1 119905) 119901(119873 119905)

just as in Section 22 but with 119905 being a continuous ratherthan a discrete variable Time-continuous strongly nonlinearMarkov processes on discrete state spaces can be describedbymeans of nonlinear master equations of the form (see eg[14])

119901 (119896 119905) = 119901 (119896 1199051015840

)

+ int

119905

1199051015840

[

119873

sum

119894=1

119908 (119894 997888rarr 119896 119904 119901 (119904)) sdot 119901 (119894 119904)

minus119901 (119896 119904)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119904 119901 (119904))] 119889119904

(17)

for 119905 gt 1199051015840 and 119896 = 1 119873 with 119908(119896 rarr 119896) = 0 for all

states 119896 Here the variables119908(119894 rarr 119896) ge 0 are transition ratesthat describe the rate of transitions (number of transitions perunit time) from states 119894 to states 119896 We will use the convention119908(119896 rarr 119896) = 0 for all states 119896 Note that as opposed to theconditional probabilities 119879(sdot) occurring in a Markov chainthe rates 119908(sdot) of the master equation are not normalizedTransition rates may depend explicitly on time such that119908(119894 rarr 119896 119905) For strongly nonlinear stochastic processestransition rates do not only depend on the states 119894 and 119896

and on time but also depend on an expectation value or theoccupation probabilities 119901(119896 119905) Note that (17) is the integral

ISRNMathematical Physics 5

form of the master equation If the integral is evaluated for asmall time interval Δ119905 = 119905 minus 119905

1015840

rarr 0 then the integral versionexhibits the structure of a Markov chain (7) with

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905)) = Δ119905 sdot 119908 (119894 997888rarr 119896 119905 119901 (119905)) 119894 = 119896

119879 (119896 997888rarr 119896 119905 Δ119905 119901 (119905)) = 1 minus Δ119905

119873

sum

119895=1

119908 (119896 997888rarr 119895 119905 119901 (119905))

(18)

which can be written in a more concise form like

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905))

= Δ119905 119908 (119894 997888rarr 119896 119905 119901 (119905)) + 120575 (119894 119896)

sdot [

[

1 minus Δ119905

119873

sum

119895=1

119908 (119894 997888rarr 119895 119905 119901 (119905))]

]

(19)

Note again that we will use the convention that the diagonalelements 119908(119896 rarr 119896) of the transition rates equal zero whichimplies that for the diagonal elements 119879(119896 rarr 119896) only thesecond term in (19) matters as stated in (18) For the sake ofcompleteness note that the differential version of the masterequation (17) reads

119889

119889119905119901 (119896 119905) =

119873

sum

119894=1

119908 (119894 997888rarr 119896 119905 119901 (119905)) 119901 (119894 119905)

minus 119901 (119896 119905)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119905 119901 (119905))

(20)

Just as in Section 22 in order to describe the four classes ofstochastic processes listed in Table 1 on an equal footing weconsider an alternative description of the process defined by(20) Accordingly we introduce the flow functionΦ(119905 120576) thatmaps initial states119883(0) defined on the integer set 1 119873 tostates119883(119905)

119883 (119905) = Φ (119883 (0) 120576) (21)

The variable 120576 is a random variable that is uniformly dis-tributed on the interval [0 1] at every time point 119905 and isstatistically independently distributed across time In view ofthe fact that the time-discrete version of the master equationexhibits the structure of a Markov chain trajectories definedby the flow function can be regarded as time-continuouscounterparts to the trajectories of Markov chains as definedby (1) and (8)ndash(11)This analogy can be exploited to constructiteratively the flow function Φ in the limiting case Δ119905 rarr 0

like

Φ 119883 (119905 + Δ119905) = 119891 (119883 (119905) 119905 Δ119905 119901 (119905) 120576 (119905)) (22)

with

119891 (119883 (119905) = 119894 119905 Δ119905 119901 (119905) 120576 (119905))

=

119873

sum

119896=1

119896 sdot 119868 (119894 997888rarr 119896 119905 Δ119905 119901 (119905) 120576 (119905))

(23)

The iterative map 119891(sdot) involves the indicator function

119868 (119894 997888rarr 119896 120576 119905 Δ119905 119901) = 1 for 120576 isin [119878 (119894 119896 minus 1) 119878 (119894 119896))

0 otherwise(24)

and the cumulative transition probabilities

119878 (119894 119896) =

119896

sum

119895=1

119879 (119894 997888rarr 119895 119905 Δ119905 119901 (119905)) (25)

Equations (1) (18) and (22)ndash(25) provide a closed approxi-mate description of the trajectories of interest By definitionthe description becomes exact in the limiting case of infinitelysmall time intervals Δ119905 Note that the conditional (transition)probabilities 119879(sdot) defined by (18) are in the interval [0 1]provided that Δ119905 is chosen sufficiently small In the limitingcase Δ119905 rarr 0 this is always the case assuming that the rates119908(sdot) are bounded from above Note also that the probabilities119879(sdot) defined by (18) are normalized to 1 Furthermore from(18) and (22)ndash(25) it follows that 119891(sdot) = 119894 holds for Δ119905 = 0That is the iterativemap (22) describes transitions from states119883(119905) = 119894 to states119883(119905+Δ119905) in the small time intervalΔ119905 Mostimportantly according to (22) the evolution of realizationsdepends on the occupation probability vector 119901(119905) at everytime point 119905 Therefore processes defined by (1) (18) and(22)ndash(25) belong to the class of strongly nonlinear stochasticprocesses (see the definition in Section 12)

25 Strongly Nonlinear Markov Diffusion Processes Defined byStrongly Nonlinear Langevin Equations and Strongly Nonline-ar Fokker-Planck Equations Realizations of time-continuousstochastic processes evolving on119873-dimensional continuousstate spaces are described by time-dependent state vectors119883(119905) with time-dependent components 119883

1(119905) 119883

119873(119905) At

every time point 119905 the vector 119883(119905) is distributed accordingto the probability density 119875(119909 119905) defined by (12) where thevariable 119909 is the state vector with coordinates 119909

1 119909

119873

Strongly nonlinear Markov diffusion processes have realiza-tions (or trajectories) defined in the limiting case of infinitelysmall time intervals Δ119905 rarr 0 by strongly nonlinear stochasticdifference equations [9] of the form

119883 (119905 + Δ119905) = 119883 (119905) + Δ119905ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ radic2Δ119905119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(26)

Carrying out the limiting procedure (26) can be writtenin a more concise way in terms of a so-called Ito-Langevinequation

119889

119889119905119883 (119905) = ℎ (119883 (119905) 119905 120579

119883(119905)[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot Γ (119905)

(27)

Just as in the case of the stochastic iterative maps discussed inSection 23 (26) and (27) exhibit two different components

6 ISRNMathematical Physics

First the vector-valued function ℎ(sdot) occurring in (26) and(27) is the drift function introduced in Section 23 thataccounts for the deterministic evolution of a process Secondthe expressions 119892 sdot 120576 and 119892 sdot Γ respectively describe avector-valued fluctuating force The variable 120576(119905) is the 119873-dimensional random variable introduced in Section 23 Thefunction Γ(119905) in the Ito-Langevin equation (27) is the coun-terpart to the random vector 120576(119905) in (26) and corresponds to aso-called vector-valued Langevin force [4 6] More preciselyeach component Γ

119896(119905) is a Langevin force We will consider

Langevin forces normalized with respect to a magnitude of 2like

lim119877rarrinfin

1

119877Γ119903

119895(119905) Γ119903

119896(119904) = 2120575 (119895 119896) 120575 (119905 minus 119904) (28)

where Γ119903

119896(119905) are realizations of the 119896th component The

variable 119892 is the 119873 times 119873 weight matrix introduced inSection 23

The stochastic process defined by the Ito-Langevin equa-tion (27) can equivalently be expressed bymeans of a stronglynonlinear Fokker-Planck equation

120597

120597119905119875 (119909 119905) = minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119875 (119909 119905)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119875 (119909 119905)]

(29)

The evolution equation for 119875(119909 119905) is nonlinear for 119875(119909 119905)In contrast the corresponding evolution equation for theconditional probability density 119879(119909 119905 | 119910 119904) with 119905 gt 119904 islinear with respect to 119879(119909 119905 | 119910 119904) and reads

120597

120597119905119879 (119909 119905 | 119910 119904)

= minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119879 (119909 119905 | 119910 119904)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119879 (119909 119905 | 119910 119904)]

(30)

For details see Frank [9] A complete description of thestochastic process can be obtained either from the realiza-tions defined by (26) and (27) or from the evolution equations(29) and (30) for 119875(119909 119905) and 119879(119909 119905 | 119910 119904) The expression

119863119894119895(119909 119905 120579

119909[119875 (sdot 119905)]) =

119873

sum

119896=1

119892119894119896(sdot) 119892119895119896(sdot) (31)

occurring in (29) and (30) is the diffusion coefficient of theprocess

A fundamental example of a strongly nonlinear Markovdiffusion process is a generalized Ornstein-Uhlenbeck pro-cess involving a mean field force ℎMF proportional to the

mean ⟨119883(119905)⟩ of the process [9 15 16] For 119873 = 1 thestochastic difference equation reads

119883 (119905 + Δ119905) = 119883 (119905) minus Δ119905 [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + radic2Δ119905119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(32)

with 119892 ge 0 Typically the case119860 119861 gt 0 is considered For (32)the strongly nonlinear Langevin equation reads

119889

119889119905119883 (119905) = minus [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + 119892Γ (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(33)

Finally the corresponding strongly nonlinear Fokker-Planckequation reads

120597

120597119905119875 (119909 119905) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119875 (119909 119905)]

+ 1198631205972

1205971199092119875 (119909 119905)

120597

120597119905119879 (119909 119905 | 119910 119904) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119879 (119909 119905 | 119910 119904)]

+ 1198631205972

1205971199092119879 (119909 119905 | 119910 119904)

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

119863 = 1198922

(34)

The generalized Ornstein-Uhlenbeck process in its form asstochastic difference equation (32) corresponds to a specialcase of the generalized autoregressive model of order 1defined by (16) Model (34) has been studied in the contextof the Shimizu-Yamada model We will return to this issuelater

26 On a General Stochastic Evolution Equation for StronglyNonlinear Markov Processes In the previous sections weconsidered some benchmark examples of strongly nonlinearstochastic processes belonging to the four classes of processesdefined in Table 1 All processes considered in Sections 22to 25 satisfy the Markov property A more general formaldescription of strongly nonlinear Markov processes thatholds for all four classes reads

119883 (119905 + 120585) = 119891 (119883 (119905) 119905 120585 120579119883(119905)

[119875 (119909 119905)] 120576 (119905)) (35)

The additional variable 120585 defines the characteristic jumpinginterval of the process under consideration and can be used todistinguish between time-discrete and time-continuous pro-cesses For 120585 = 1 we are dealing with time-discrete stochastic

ISRNMathematical Physics 7

processes In the case of processes defined on discrete statespaces 119875(119909 119905) is related to the occupation probability 119901(119896 119905)by means of (5) and (6) and (35) reduces to the special caseof (8) of the Markov chain model For processes defined oncontinuous state spaces and 120585 = 1 the general form (35)becomes (13) in the case of stochastic processes driven bynonparametric white noise In the limiting case 120585 rarr 0 weare dealing with time-continuous stochastic processes In thiscase the function 119891(sdot) possesses the property 119891(sdot) = 119883(119905)

for 120585 = 0 For processes that evolve on discrete state spacesand satisfy a nonlinear master equation it follows that (35)becomes the iterative map (22) provided that the argument119875(119909 119905) in 119891(sdot) is expressed in terms of the occupationprobability 119901(119896 119905) see (6) In contrast for processes evolvingon continuous state spaces and satisfying strongly nonlinearIto-Langevin equations the general form (35) reduces to thestochastic difference equation (26) when we consider thelimiting case 120585 rarr 0

3 Time-Discrete Strongly Nonlinear StochasticProcesses on Discrete State Spaces

31 The Strongly Nonlinear Two-State Markov Chain ModelA fundamental strongly nonlinear stochastic process is thetwo-state nonlinear Markov chain process [17] For the sakeof convenience the states 119896 = 1 and 119896 = 2will be labeled withldquo119860rdquo and ldquo119861rdquo Transitions from119860 to119860119860 to 119861 119861 to119860 and 119861 to119861 occur with conditional probabilities 119879(119860 rarr 119860) 119879(119860 rarr

119861) 119879(119861 rarr 119860) and 119879(119861 rarr 119861) respectively Accordinglythe nonlinear Markov chain equation (7) reads

119901 (119860 119905 + 1) = 119879 (119860 997888rarr 119860 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119905)) 119901 (119861 119905)

119901 (119861 119905 + 1) = 119879 (119860 997888rarr 119861 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119861 119905 119901 (119905)) 119901 (119861 119905)

(36)

Taking the normalization condition 119901(119860) + 119901(119861) = 1 intoconsideration (36) can equivalently be expressed by

119901 (119860 119905 + 1)

= [119879 (119860 997888rarr 119860 119905 119901 (119860 119905)) minus 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))] 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))

(37)

which is a closed nonlinear equation in 119901(119860 119905) Model (37)involves two functions only the conditional probabilities119879(119860 rarr 119860) and 119879(119861 rarr 119860) which can be chosenindependently from each other and in the case of a nonlinearMarkov process depend on 119901(119860 119905) as indicated in (37)The remaining conditional probabilities can be calculatedfrom 119879(119860 rarr 119861) = 1 minus 119879(119860 rarr 119860) and 119879(119861 rarr

119861) = 1 minus 119879(119861 rarr 119860) Model (37) has been examined inphysics social sciences and psychology as will be discussednext

32 Chaos in Probability Time-Discrete Case FollowingFrank [18] let us put 119879(119860 rarr 119860) = 119879(119861 rarr 119860) = 119879

0 Then

(37) becomes

119901 (119860 119905 + 1) = 1198790(119905 119901 (119860 119905)) (38)

Although (38) describes a stochastic process formally (38)exhibits the structure of a nonlinear deterministic mapsubjected to the constraint that the function 119879

0must be in

the interval [0 1] at any time step 119905 and for any occupationprobability 119901(119860) Iterative maps in general are known toexhibit chaos [19] Therefore (38) suggests that time-discretestrongly nonlinear Markov processes exhibit chaos as wellSuch strongly nonlinear chaotic processes exhibit chaos in theevolution of the occupation probabilities and the conditionalprobabilities For example in Frank [18] the logistic function1198790(119911) = 120572 sdot 119911(1 minus 119911) [19 20] was considered for which (38)

reads

119901 (119860 119905 + 1) = 120572119901 (119860 119905) [1 minus 119901 (119860 119905)] (39)

For 120572 in [0 4] the function 1198790(sdot) is in the interval [0 1] such

that (39) maps the occupation probability 119901(119860) from theinterval [0 1] to the interval [0 1] as required For parameters120572 ge 120572crit asymp 3570 [20] the map exhibits chaotic solutionsThat is 119901(119860 119905) evolves in a chaotic fashion However asstated previously not only the occupation probability exhibitschaos but the conditional probabilities 119879(119860 rarr 119860) =

119879(119861 rarr 119860) = 1198790exhibit a chaotic behavior as well which

was demonstrated explicitly by numerical simulations [18]Chaos in strongly nonlinear time-discrete Markov processesis not limited to the two-state models In fact the two-statemodel can be generalized to an 119873-state model In this case(38) becomes [18]

119901 (119896 119905 + 1) = 1198790(119905 119901 (119905)) (40)

for 119896 = 1 119873 minus 1 The occupation probability 119901(119873 119905)

is computed from the normalization condition Conse-quently 119879

0depends only on the occupation probabilities

119901(1 119905) 119901(119873 minus 1 119905) For example the two-dimensionalchaotic Bakermap [20] was described bymeans of (40) using119873 = 3 [18]

33 Social Sciences andGroupDecisionMaking Time-DiscreteCase Group decision making has frequently been describedby nonlinear stochastic processes [21ndash24]The reason for thisis that under certain circumstances individuals are likely tobe affected by the opinion that is presented by a group asa whole rather than by the opinion of individuals This so-called group opinion in turn is produced by the opinions anddecisions of the individual group members Consequentlythe decision making process of an individual is affected bythe sample average obtained from the opinions of the othersIf the groups can be considered as being relatively large thesample averagemay be replaced by the ensemble averageThedecision making behavior of an individual is then consideredas a realization of a stochastic process that is affected by theensemble average of the process In the context of votingbehavior the two-state model introduced in Section 31 was

8 ISRNMathematical Physics

applied [17]Themembers of theUSHouse of Representativeshad the choice to vote for or against passing the so-calledbailout bill (US House of Representatives USA 2008) Thebill failed to pass in the September 29 vote In a second voteon October 3 the bill finally passedThat is a critical numberof ldquoNordquo voters changed their opinions between September 29and October 3 The states 119896 = 1 and 119896 = 2 were labeled byldquo119884rdquo and ldquo119873rdquo representing ldquoYesrdquo and ldquoNordquo voters Accordingto model (37) the conditional probabilities 119879(119873 rarr 119884) and119879(119884 rarr 119884) describing a change in the opinion from ldquoNordquo toldquoYesrdquo and a consistent ldquoYesrdquo opinion were considered Usingthe notation of (37) the voting model proposed in Frank [18]reads

119901 (119884 119905 + 1)

= [119879 (119884 997888rarr 119884 119905 119901 (119884 119905)) minus 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))] 119901 (119884 119905)

+ 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))

(41)

The data available in the literature suggested that 119879(119884 rarr

119884) = 1 For the119879(119873 rarr 119884) the function119879(119873 rarr 119884) = 119860minus119861sdot

119901(119884 119905) was used to describe the impact of the group opinionon the voting behavior This relationship assumes that thepressure to change from a ldquoNordquo voter to a ldquoYesrdquo voter was highimmediately subsequent to the September 29 votes when theprobability 119901(119884 119905) at 119905 = Sept 29 was not high enough tomakethe bill pass In this context it should be mentioned that thefailure of the bill to pass on September 29 triggered a dramaticbreakdown of the US stock market That is the members ofthe House of Representatives who had voted with ldquoNordquo werenot only subjected to political pressure but also experiencedessential pressure from the economy For 119879(119884 rarr 119884) = 1

and 119879(119873 rarr 119884) = 119860 minus 119861119901(119884 119905) the stationary occupationprobability 119901(119884 st) is given by 119901(119884 st) = 119860119861 such that (41)eventually reads

119901 (119884 119905 + 1) = 119901 (119884 119905) + 119861 [119901 (119884 st) minus 119901 (119884 119905)] [1 minus 119901 (119884 119905)]

(42)

The stationary value 119901(119884 st) was estimated from the dataA detailed model-based analysis of the data showed thatthe members of the two main parties the Democratic andRepublican members were differently affected by the grouppressure

34 Psychology and Human Memory A pioneering experi-ment in humanmemory research is the free recall experimentin which participants are asked to recall items of a particularcategory [25 26] For example they are asked to recall animalnames Typically participants are instructed to recall as manyitems as possible and to do so as fast as possible and withoutrepetitionThe total number of recalled items is by definitiona monotonically increasing function of time Strikinglyfree recall involves clusters in which item subcategories arerecalled more or less as a whole For example when recallinganimal names a participant may recall a number of insectnames in succession From a theoretical point of view at anygiven time point during the free recall process the ability

to recall items in clusters depends on the size of the poolof items available for recalling In other words if there isa large pool of items that have not yet been recalled thenthere is a high chance that a whole item subcategory canbe recalled In line with this argument a stochastic modelfor free recall dynamics was developed in which the recallprocess depends on the size of the pool of items availablefor recall [27] More precisely time was parceled in discretetime intervals in which recall attempts are executed The aimwas to develop a model for the probability that a single itemhas been or has not yet been recalled at a given time step 119905To this end the two-state model of Section 31 was appliedAccordingly the states 119896 = 1 and 119896 = 2 correspond tobeing recalled or being not yet been recalled and are labeledby ldquo119877rdquo and ldquo119873rdquo respectively The two relevant conditionalprobabilities are119879(119877 rarr 119877) and119879(119873 rarr 119877) By the design ofthe experiment we have119879(119877 rarr 119877) = 1 Moreover as arguedpreviously if the pool of items that have not yet recalled islarge that is if the probability 119901(119873) that an item has notyet been recalled is high then the conditional probability119879(119873 rarr 119877) should be high aswell Likewise if the probability119901(119877) that an item has been recalled is high then 119879(119873 rarr 119877)

should be relatively low Therefore it was suggested to put119879(119873 rarr 119877) = 119861 sdot (1 minus 119901(119877 119905)) with 119861 gt 0 This functionimplies that the occupation probability 119901(119877 119905) converges tothe stationary fixed 119901(119877 st) = 1 at which 119879(119873 rarr 119877) = 0The full model can be obtained by substituting 119879(119877 rarr 119877)

and 119879(119873 rarr 119877) into (37) and reads

119901 (119877 119905 + 1) = 119901 (119877 119905) + 119861[1 minus 119901 (119877 119905)]2

(43)

Note that from a mathematical point of view this modelis a special case of model (42) for voting behavior (hintput 119901(119884 st) = 1 in (42)) Since the model describes thesingle item recall probability the model must be scaled tothe number of items recalled by any given participant Tothis end a parameter 119862 describing the maximal number ofitems for recall was introduced The unknown parameters119861 and 119862 were estimated for free recall data collected in astudy by Rhodes and Turvey [28] and the sample meanvalues 119861 = 0009 (SD = 0005) and 119862 = 307 (SD = 138)were found for a sample of eight participants [27] Note that(43) predicts that the fixed point 119901(119877 st) = 1 will neverbe reached in a finite number of time steps Consequentlythe model predicts that the total number of recalled items119872 at the end of the experiment is smaller than the numberof available items for recall This prediction was confirmedby the data Moreover the model-based analysis revealed astatistically significant positive correlation between 119872 and119862 for the participant sample studied in Frank and Rhodes[27]

35 Biology and Evolutionary Population Dynamics A pri-mary concern of evolutionary population dynamics is theunderstanding of the origin of life Among the many modelsthat have been proposed in this context the studies byCrutchfield and Gornerup [29] and Gornerup and Crutch-field [30] are of interest for our review In these studiesa number of 119873 macromolecules with different functional

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Stochastic AnalysisInternational Journal of

Page 2: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

2 ISRNMathematical Physics

two alternative terms have been proposed in the literatureWehner and Wolfer [8] suggested the term ldquotruly nonlinearFokker-Planck equationsrdquo whereas Frank [9] coined the termldquostrongly nonlinear Fokker-Planck equationsrdquo Only recentlyit has become clear that the class of stochastic processesaddressed by McKean is not limited to be described by drift-diffusion equations of the Fokker-Planck type Thereforethe research field inspired by the work of McKean is aboutnonlinear stochastic processes in general To give processes ofthis kind amore specific name wemay adopt the suggestionsby Wehner and Wolfer or Frank and say that there is a novelclass of processes called ldquotruly nonlinear stochastic processesrdquoor ldquostrongly nonlinear stochastic processesrdquo In what followswe will use the term ldquostronglyrdquo that should be considered asa synonym to ldquotrulyrdquo just to avoid repeating the phrase ldquotrulyor stronglyrdquo over and over again

For the purpose of this paper the following unifying def-inition of a strongly nonlinear process will be used

A strongly nonlinear stochastic process is a sto-chastic process for which the evolution of realiza-tions depends on at least one expectation valueof the process or on the appropriately definedprobability density of that process

Note that for strongly nonlinear processes evolving on dis-crete state spaces the realizations are required to depend on atleast one expectation value or on the occupation probabilityrather than the probability density However as we willdemonstrate in Section 2 for our purposes the occupationprobability of a discrete-state process can be derived froman appropriately defined probability density Therefore thedefinition as given above holds both for processes evolvingon continuous and discrete state spaces

In this review of strongly nonlinear stochastic processesa distinction will be made between four classes of processesThese classes arise naturally if it is taken into considerationthat in many disciplines time is not necessarily considered tobe a continuous variable Rather time might be consideredas a discrete variable such that processes evolve in discretetime steps For example a process describing the evolutionof a stock price on a day-to-day basis is typically modelledby means of a time-discrete process [10 11] In short timecan be a discrete or continuous variable Likewise the statespace of a process might be discrete or continuous The fourpossible combinations of time and space characteristics giverise to four classes of stochastic processes We will addressthese classes in separate sections see Table 1

Before we proceed in Sections 3 4 5 and 6 with thesystematic presentation outlined in Table 1 we will presentin Section 2 benchmark examples of processes for each of thefour classes For the examples discussed in Section 2 a unify-ing mathematical definition irrespective of class membershipwill be presented in the final subsection of Section 2 Meanfield theory and themathematical literature will be addressedin Sections 7 and 8 As it turns out all examples reviewedin Sections 2 to 6 exhibit the Markov property ThereforeSection 9 is devoted to the review of somemore recent studieson non-Markovian strongly nonlinear stochastic processes

Table 1 Four classes of strongly nonlinear stochastic processes andsections in which they will be addressed

Time discrete Time continuousSpace discrete Section 3 Section 5Space continuous Section 4 Section 6

In Section 10 some key issues of strongly nonlinear stochasticprocesses will be highlighted

2 Benchmark Examples of Strongly NonlinearProcesses and on a General StochasticEvolution Equation for a Broad Class ofStrongly Nonlinear Markov Processes

21 Some Notations At the present stage it is useful tointroduce some notations in order to give some benchmarkexamples for each of the four classes indicated in Table 1

Time-discrete processes evolve on discrete time steps119905 = 1 2 3 In contrast time-continuous processes aredescribed by a continuous time variable 119905 defined on someinterval For our purposes this interval will go from zero toinfinity meaning that we will consider initial value problemsstarting at an initial time 119905 = 0 and extending in timeinfinitely Note that the symbol 119905 will be used to denote bothdiscrete-valued and continuous time variables

Processes evolving in discrete finite state spaces exhibit anumber of 119873 states labeled by 119896 = 1 2 119873 Let 119883 denotethe state variable of a state discrete process which impliesthat 119883 assumes an integer between 1 and 119873 Moreover let119883119903 denote the 119903th realization of the process A fundamental

(although not complete) stochastic description of processesevolving on finite discrete state spaces is given by theoccupation probabilities 119901(119896) The occupation probability119901(119896) defines the probability that the state 119896 is occupied It canbe computed from a number of 119877 realizations in the limitingcase of 119877 rarr infin like

119901 (119896) = lim119877rarrinfin

1

119877120575 (119883119903

119896) (1)

The function120575(sdot) is theKronecker functionwhich equals 1 for119883119903

= 119896 and equals 0 otherwiseThe probabilities119901(119896) definedby (1) satisfy the normalization condition

119873

sum

119896=1

119901 (119896) = 1 (2)

Processes evolving on finite dimensional continuousstate spaces can be described by a state vector 119883 withcoefficients 119883

1 119883

119873 The elements or coordinates 119883

119896are

defined on certain domains of definition Ω119896 For example

a domain of definition may correspond to the real lineAn important measure characterizing stochastic processesevolving on continuous state spaces is the probability density119875 defined as the ensemble average over the Dirac deltafunction 120575(sdot) Note that the same symbol 120575(sdot) will be used forthe Kronecker function and the Dirac delta function in order

ISRNMathematical Physics 3

to stress the analogies between the classes listed in Table 1Mathematically speaking the probability density 119875 can bedefined by

119875 (119909) = lim119877rarrinfin

1

119877120575 (119883119903

1minus 1199091) sdot sdot sdot 120575 (119883

119903

119873minus 119909119873) (3)

In (3) the variable 119909 is the state vector with coordinates1199091 119909

119873 whereas the elements119883119903

1 119883

119903

119873are the elements

of the 119903th vector-valued realization 119883119903 of the process The

probability density satisfies the normalization condition

int

Ωtot

119875 (119909)

119873

prod

119896=1

119889119909119896= 1 Ωtot =

119873

prod

119896=1

Ω119896 (4)

where Ω119896are the domains of definitions mentioned previ-

ouslyLet us return to processes defined on a discrete state space

In addition we put119873 = 1 and assume thatΩtot = Ω1denotes

the real half-line from 0 to infinity Formally the probabilitydensity 119875(119909) assumes the form

119875 (119909) =

119873

sum

119896=1

119901 (119896) 120575 (119909 minus 119896) (5)

where 119901(119896) are the occupation probabilities of the states 119896defined in (1) In turn the occupation probabilities can becalculated from 119875(119909) like

119901 (119896) = int

119896+120576

119896minus120576

119875 (119909) 119889119909 (6)

where 120576 is a positive number smaller than 1 (eg 120576 = 01)In view of (6) the occupation probabilities 119901(119896)may be con-sidered as functions of an appropriately defined probabilitydensity119875 as stated in Section 1With these notations at handsome benchmark examples of strongly nonlinear stochasticprocesses can be given

22 Strongly Nonlinear Markov Processes Described by Non-linear Markov Chains In general the occupation proba-bilities 119901(119896) of time-discrete stochastic processes evolvingon discrete state spaces constitute a time-dependent vector119901(119905) with coefficients 119901(1 119905) 119901(119873 119905) For the importantspecial case of nonlinear Markov processes described bynonlinearMarkov chains the occupation probabilities satisfy[12]

119901 (119896 119905 + 1) =

119873

sum

119894=1

119879 (119894 997888rarr 119896 119905 119901 (119905)) 119901 (119894 119905) (7)

Here 119879 describes the transition probability matrix of theprocess The matrix is an119873 times119873matrix with coefficients119879(119894 rarr 119896) The coefficients 119879(119894 rarr 119896) describe theconditional probabilities that transitions from state 119894 to state119896 occur given that the process is in the state 119894The conditionalprobabilities 119879(119894 rarr 119896)may depend explicitly on time 119905 suchthat 119879(119894 rarr 119896 119905) A nonlinear Markov process is character-ized by the property that at least one conditional probability

depends at least on one expectation value computed from theprocess at time step 119905 or on one occupation probability119901(119896 119905)In this case we have 119879(119894 rarr 119896 119905 119901(119905)) as indicated in (7) Acomplete description of the stochastic process can be derivedfrom (7) as shown in Frank [12] In view of the definition ofa strongly nonlinear stochastic process given in Section 12that is based on realizations of stochastic processes it is worthwhile to consider an alternative definition of the process Tothis end the stochastic map 119891(sdot 120576) can be defined that mapsa state119883(119905) to a state119883(119905 + 1) like

119883 (119905 + 1) = 119891 (119883 (119905) 119905 119901 (119905) 120576 (119905)) (8)

The mapping 119891(sdot 120576) depends on a random variable 120576(119905) thatis uniformly distributed on the interval [0 1] at any timestep 119905 Across time 120576(119905) is statistically independent Thestochastic map 119891(sdot 120576) describes the jumps from119883(119905) to119883(119905+1) satisfying the transition probabilities 119879(119894 rarr 119896) In orderto derive an explicit definition of 119891(sdot 120576) we first note thatthe conditional probabilities 119879(119894 rarr 119896) are by definition realnumbers between 0 and 1 that add up to unity Next it ususeful to consider the cumulative conditional probabilities119878(119894 119896) defined for 119894 = 1 2 119873 and 119896 = 0 1 119873 by

119878 (119894 119896) =

119896

sum

119895=1

119879 (119894 997888rarr 119895 119905 119901 (119905)) (9)

for 119896 gt 0 For 119896 = 0 we put 119878(119894 0) = 0 For 119896 gt 0

the cumulativemeasures 119878(119894 119896) describe the probabilities thattransitions from state 119894 to one of the states 1 2 119896 occur attime 119905 given that the process has an occupation probabilityvector 119901 at time 119905 The cumulative measures represent aset of monotonically increasing points on the interval [0 1]such that the interval between two adjacent points 119878(119894 119896 minus

1) and 119878(119894 119896) equals 119879(119894 rarr 119896) Consequently the pointspectrum 119878(119894 119896) in combinationwith the randomvariable 120576(119905)can be used to construct realizations of the process underconsideration If 119883(119905) = 119894 holds and if 120576(119905) falls into theinterval [119878(119894 119896 minus 1) 119878(119894 119896)) then we put119883(119905+ 1) = 119896 If we doso then transitions from 119894 to 119896 happen with the probability119879(119894 rarr 119896) if the process is in state 119894 at 119905 and distributed like119901(119905) at 119905mdashas required A concise mathematical formulationof this idea can be obtained with the help of the indicatorfunction 119868(119894 rarr 119896 120576) defined by

119868 (119894 997888rarr 119896 119905 119901 120576) = 1 for 120576 isin [119878 (119894 119896 minus 1) 119878 (119894 119896))

0 otherwise(10)

Then the map 119891(sdot 120576) can be defined as follows

119891 (119883 (119905) = 119894 119905 119901 (119905) 120576 (119905)) =

119873

sum

119896=1

119896 sdot 119868 (119894 997888rarr 119896 119905 119901 (119905) 120576 (119905))

(11)

As indicated in (11) the mapping depends on the probabilityvector 119901(119905) because the indicator function 119868(sdot) depends on119901(119905) via the conditional probabilities 119878 and 119879 Thereforeeach realization 119883

119903

(119905) of the process defined by (8)ndash(11)depends on the occupation probabilities119901(119896 119905) of the process

4 ISRNMathematical Physics

The occupation probabilities in turn can be computed fromthe realizations see (1) Therefore the mathematical modelgiven by (1) and (8)ndash(11) provides a closed (and complete)description for the strongly nonlinear stochastic processdescribed by the nonlinear Markov chain (7)

23 Strongly Nonlinear Markov Processes Described by Sto-chastic Iterative Maps The realization of a time-discrete sto-chastic process on an119873-dimensional continuous state spaceis described by a time-dependent state vector119883(119905)with time-dependent components119883

1(119905) 119883

119873(119905) From (3) it follows

that at any time step 119905 the vector119883(119905) is distributed accordingto the time-dependent probability density

119875 (119909 119905) = lim119877rarrinfin

1

119877120575 (119883119903

1(119905) minus 119909

1) sdot sdot sdot 120575 (119883

119903

119873(119905) minus 119909

119873)

(12)

Strongly nonlinear Markov processes defined by stochasticiterative maps with nonparametric white noise satisfy equa-tions of the general form

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(13)

A simplified version of (13) was introduced earlier in Frank[13] In (13) the vector-valued variable ℎ(sdot) is called the driftfunction and describes the deterministic evolution of theprocess Second the term 119892(sdot) sdot 120576(119905) in (13) describes thevector-valued fluctuating force of the stochastic process Therandom vector 120576(119905) with coefficients 120576

1(119905) 120576

119873(119905) is an 119873-

dimensional normally distributed white noise process Thatis each coefficient 120576

119896(119905) is normally distributed at any time

step 119905 and statistically independent in time The randomcoefficients 120576

119896(119905) are statistically independent among each

otherThe variable 119892 is an119873 times119873matrix with coefficients119892119895119896 The coefficients 119892

119895119896with 119896 = 1 119873 for a fixed index

119895 weight the contributions that the random components1205761(119905) 120576

119873(119905) have on the evolution of the process in the

direction (or along the coordinate) 119909119895 In particular for

119892119895119896(sdot 119905) = 0 the random coefficient 120576

119896(119905) does not occur

in the evolution equation of 119883119895at time 119905 Therefore 119892 may

be considered as weight matrix Importantly the coefficients119892119895119896

measure the strength of the fluctuating force 119892(sdot) sdot 120576(119905)of the stochastic process described by (13) The reason forthis is that the random vector 120576(119905) is normalized at any timestep 119905 Most importantly in the case of a strongly nonlinearMarkov process the drift function ℎ(sdot) or the fluctuating force119892(sdot) sdot 120576(119905) or both depend on at least one expectation value oron the probability density 119875 as indicated in (13)The operator120579 maps the probability density 119875 to a real number or a set ofreal numbers This mapping may depend on the value of 119883For example the probability density may be evaluated at thevalue of the process like

120579119883(119905)

[119875 (119909 119905)] = 119875 (119909 119905)|119909=119883(119905)

(14)

In this context it should be noted that the formal depen-dencies ℎ(120579

119883[119875]) and 119892(120579

119883[119875]) include the dependencies on

expectation values (such as the mean) as special cases Forexample

120579119883(119905)

[119875 (119909 119905)] = int

Ωtot

119909119875 (119909 119905)

119873

prod

119896=1

119889119909119896= ⟨119883 (119905)⟩ (15)

where ⟨sdot⟩ is the operator symbol for an ensemble average suchthat ⟨119883⟩ denotes the mean of the random vector 119883 Notethat (12) and (13) provide a closed description of a stochasticprocess Moreover from the structure of (13) it followsimmediately that processes defined by (12) and (13) satisfythe definition of strongly nonlinear stochastic processes pre-sented in Section 12 An illustrative example of a stochasticiterative map (13) is the autoregressive model of order 1subjected to a so-called mean field force ℎMF proportional tothe mean ⟨119883(119905)⟩ of the process For 119873 = 1 the model reads[13]

119883 (119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(16)

with 119892 ge 0 where the parameters119860 and 119861 are scalarsWe willreturn to model (16) in Section 4

24 Strongly Nonlinear Markov Processes Described by Non-linearMaster Equations For time-continuous stochastic pro-cesses evolving on discrete state spaces the occupation prob-abilities 119901(119896) are functions of time 119905 where 119905 is a continuousvariable Likewise the probability vector 119901(119905) becomes atime-dependent variable with coefficients 119901(1 119905) 119901(119873 119905)

just as in Section 22 but with 119905 being a continuous ratherthan a discrete variable Time-continuous strongly nonlinearMarkov processes on discrete state spaces can be describedbymeans of nonlinear master equations of the form (see eg[14])

119901 (119896 119905) = 119901 (119896 1199051015840

)

+ int

119905

1199051015840

[

119873

sum

119894=1

119908 (119894 997888rarr 119896 119904 119901 (119904)) sdot 119901 (119894 119904)

minus119901 (119896 119904)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119904 119901 (119904))] 119889119904

(17)

for 119905 gt 1199051015840 and 119896 = 1 119873 with 119908(119896 rarr 119896) = 0 for all

states 119896 Here the variables119908(119894 rarr 119896) ge 0 are transition ratesthat describe the rate of transitions (number of transitions perunit time) from states 119894 to states 119896 We will use the convention119908(119896 rarr 119896) = 0 for all states 119896 Note that as opposed to theconditional probabilities 119879(sdot) occurring in a Markov chainthe rates 119908(sdot) of the master equation are not normalizedTransition rates may depend explicitly on time such that119908(119894 rarr 119896 119905) For strongly nonlinear stochastic processestransition rates do not only depend on the states 119894 and 119896

and on time but also depend on an expectation value or theoccupation probabilities 119901(119896 119905) Note that (17) is the integral

ISRNMathematical Physics 5

form of the master equation If the integral is evaluated for asmall time interval Δ119905 = 119905 minus 119905

1015840

rarr 0 then the integral versionexhibits the structure of a Markov chain (7) with

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905)) = Δ119905 sdot 119908 (119894 997888rarr 119896 119905 119901 (119905)) 119894 = 119896

119879 (119896 997888rarr 119896 119905 Δ119905 119901 (119905)) = 1 minus Δ119905

119873

sum

119895=1

119908 (119896 997888rarr 119895 119905 119901 (119905))

(18)

which can be written in a more concise form like

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905))

= Δ119905 119908 (119894 997888rarr 119896 119905 119901 (119905)) + 120575 (119894 119896)

sdot [

[

1 minus Δ119905

119873

sum

119895=1

119908 (119894 997888rarr 119895 119905 119901 (119905))]

]

(19)

Note again that we will use the convention that the diagonalelements 119908(119896 rarr 119896) of the transition rates equal zero whichimplies that for the diagonal elements 119879(119896 rarr 119896) only thesecond term in (19) matters as stated in (18) For the sake ofcompleteness note that the differential version of the masterequation (17) reads

119889

119889119905119901 (119896 119905) =

119873

sum

119894=1

119908 (119894 997888rarr 119896 119905 119901 (119905)) 119901 (119894 119905)

minus 119901 (119896 119905)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119905 119901 (119905))

(20)

Just as in Section 22 in order to describe the four classes ofstochastic processes listed in Table 1 on an equal footing weconsider an alternative description of the process defined by(20) Accordingly we introduce the flow functionΦ(119905 120576) thatmaps initial states119883(0) defined on the integer set 1 119873 tostates119883(119905)

119883 (119905) = Φ (119883 (0) 120576) (21)

The variable 120576 is a random variable that is uniformly dis-tributed on the interval [0 1] at every time point 119905 and isstatistically independently distributed across time In view ofthe fact that the time-discrete version of the master equationexhibits the structure of a Markov chain trajectories definedby the flow function can be regarded as time-continuouscounterparts to the trajectories of Markov chains as definedby (1) and (8)ndash(11)This analogy can be exploited to constructiteratively the flow function Φ in the limiting case Δ119905 rarr 0

like

Φ 119883 (119905 + Δ119905) = 119891 (119883 (119905) 119905 Δ119905 119901 (119905) 120576 (119905)) (22)

with

119891 (119883 (119905) = 119894 119905 Δ119905 119901 (119905) 120576 (119905))

=

119873

sum

119896=1

119896 sdot 119868 (119894 997888rarr 119896 119905 Δ119905 119901 (119905) 120576 (119905))

(23)

The iterative map 119891(sdot) involves the indicator function

119868 (119894 997888rarr 119896 120576 119905 Δ119905 119901) = 1 for 120576 isin [119878 (119894 119896 minus 1) 119878 (119894 119896))

0 otherwise(24)

and the cumulative transition probabilities

119878 (119894 119896) =

119896

sum

119895=1

119879 (119894 997888rarr 119895 119905 Δ119905 119901 (119905)) (25)

Equations (1) (18) and (22)ndash(25) provide a closed approxi-mate description of the trajectories of interest By definitionthe description becomes exact in the limiting case of infinitelysmall time intervals Δ119905 Note that the conditional (transition)probabilities 119879(sdot) defined by (18) are in the interval [0 1]provided that Δ119905 is chosen sufficiently small In the limitingcase Δ119905 rarr 0 this is always the case assuming that the rates119908(sdot) are bounded from above Note also that the probabilities119879(sdot) defined by (18) are normalized to 1 Furthermore from(18) and (22)ndash(25) it follows that 119891(sdot) = 119894 holds for Δ119905 = 0That is the iterativemap (22) describes transitions from states119883(119905) = 119894 to states119883(119905+Δ119905) in the small time intervalΔ119905 Mostimportantly according to (22) the evolution of realizationsdepends on the occupation probability vector 119901(119905) at everytime point 119905 Therefore processes defined by (1) (18) and(22)ndash(25) belong to the class of strongly nonlinear stochasticprocesses (see the definition in Section 12)

25 Strongly Nonlinear Markov Diffusion Processes Defined byStrongly Nonlinear Langevin Equations and Strongly Nonline-ar Fokker-Planck Equations Realizations of time-continuousstochastic processes evolving on119873-dimensional continuousstate spaces are described by time-dependent state vectors119883(119905) with time-dependent components 119883

1(119905) 119883

119873(119905) At

every time point 119905 the vector 119883(119905) is distributed accordingto the probability density 119875(119909 119905) defined by (12) where thevariable 119909 is the state vector with coordinates 119909

1 119909

119873

Strongly nonlinear Markov diffusion processes have realiza-tions (or trajectories) defined in the limiting case of infinitelysmall time intervals Δ119905 rarr 0 by strongly nonlinear stochasticdifference equations [9] of the form

119883 (119905 + Δ119905) = 119883 (119905) + Δ119905ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ radic2Δ119905119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(26)

Carrying out the limiting procedure (26) can be writtenin a more concise way in terms of a so-called Ito-Langevinequation

119889

119889119905119883 (119905) = ℎ (119883 (119905) 119905 120579

119883(119905)[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot Γ (119905)

(27)

Just as in the case of the stochastic iterative maps discussed inSection 23 (26) and (27) exhibit two different components

6 ISRNMathematical Physics

First the vector-valued function ℎ(sdot) occurring in (26) and(27) is the drift function introduced in Section 23 thataccounts for the deterministic evolution of a process Secondthe expressions 119892 sdot 120576 and 119892 sdot Γ respectively describe avector-valued fluctuating force The variable 120576(119905) is the 119873-dimensional random variable introduced in Section 23 Thefunction Γ(119905) in the Ito-Langevin equation (27) is the coun-terpart to the random vector 120576(119905) in (26) and corresponds to aso-called vector-valued Langevin force [4 6] More preciselyeach component Γ

119896(119905) is a Langevin force We will consider

Langevin forces normalized with respect to a magnitude of 2like

lim119877rarrinfin

1

119877Γ119903

119895(119905) Γ119903

119896(119904) = 2120575 (119895 119896) 120575 (119905 minus 119904) (28)

where Γ119903

119896(119905) are realizations of the 119896th component The

variable 119892 is the 119873 times 119873 weight matrix introduced inSection 23

The stochastic process defined by the Ito-Langevin equa-tion (27) can equivalently be expressed bymeans of a stronglynonlinear Fokker-Planck equation

120597

120597119905119875 (119909 119905) = minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119875 (119909 119905)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119875 (119909 119905)]

(29)

The evolution equation for 119875(119909 119905) is nonlinear for 119875(119909 119905)In contrast the corresponding evolution equation for theconditional probability density 119879(119909 119905 | 119910 119904) with 119905 gt 119904 islinear with respect to 119879(119909 119905 | 119910 119904) and reads

120597

120597119905119879 (119909 119905 | 119910 119904)

= minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119879 (119909 119905 | 119910 119904)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119879 (119909 119905 | 119910 119904)]

(30)

For details see Frank [9] A complete description of thestochastic process can be obtained either from the realiza-tions defined by (26) and (27) or from the evolution equations(29) and (30) for 119875(119909 119905) and 119879(119909 119905 | 119910 119904) The expression

119863119894119895(119909 119905 120579

119909[119875 (sdot 119905)]) =

119873

sum

119896=1

119892119894119896(sdot) 119892119895119896(sdot) (31)

occurring in (29) and (30) is the diffusion coefficient of theprocess

A fundamental example of a strongly nonlinear Markovdiffusion process is a generalized Ornstein-Uhlenbeck pro-cess involving a mean field force ℎMF proportional to the

mean ⟨119883(119905)⟩ of the process [9 15 16] For 119873 = 1 thestochastic difference equation reads

119883 (119905 + Δ119905) = 119883 (119905) minus Δ119905 [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + radic2Δ119905119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(32)

with 119892 ge 0 Typically the case119860 119861 gt 0 is considered For (32)the strongly nonlinear Langevin equation reads

119889

119889119905119883 (119905) = minus [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + 119892Γ (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(33)

Finally the corresponding strongly nonlinear Fokker-Planckequation reads

120597

120597119905119875 (119909 119905) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119875 (119909 119905)]

+ 1198631205972

1205971199092119875 (119909 119905)

120597

120597119905119879 (119909 119905 | 119910 119904) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119879 (119909 119905 | 119910 119904)]

+ 1198631205972

1205971199092119879 (119909 119905 | 119910 119904)

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

119863 = 1198922

(34)

The generalized Ornstein-Uhlenbeck process in its form asstochastic difference equation (32) corresponds to a specialcase of the generalized autoregressive model of order 1defined by (16) Model (34) has been studied in the contextof the Shimizu-Yamada model We will return to this issuelater

26 On a General Stochastic Evolution Equation for StronglyNonlinear Markov Processes In the previous sections weconsidered some benchmark examples of strongly nonlinearstochastic processes belonging to the four classes of processesdefined in Table 1 All processes considered in Sections 22to 25 satisfy the Markov property A more general formaldescription of strongly nonlinear Markov processes thatholds for all four classes reads

119883 (119905 + 120585) = 119891 (119883 (119905) 119905 120585 120579119883(119905)

[119875 (119909 119905)] 120576 (119905)) (35)

The additional variable 120585 defines the characteristic jumpinginterval of the process under consideration and can be used todistinguish between time-discrete and time-continuous pro-cesses For 120585 = 1 we are dealing with time-discrete stochastic

ISRNMathematical Physics 7

processes In the case of processes defined on discrete statespaces 119875(119909 119905) is related to the occupation probability 119901(119896 119905)by means of (5) and (6) and (35) reduces to the special caseof (8) of the Markov chain model For processes defined oncontinuous state spaces and 120585 = 1 the general form (35)becomes (13) in the case of stochastic processes driven bynonparametric white noise In the limiting case 120585 rarr 0 weare dealing with time-continuous stochastic processes In thiscase the function 119891(sdot) possesses the property 119891(sdot) = 119883(119905)

for 120585 = 0 For processes that evolve on discrete state spacesand satisfy a nonlinear master equation it follows that (35)becomes the iterative map (22) provided that the argument119875(119909 119905) in 119891(sdot) is expressed in terms of the occupationprobability 119901(119896 119905) see (6) In contrast for processes evolvingon continuous state spaces and satisfying strongly nonlinearIto-Langevin equations the general form (35) reduces to thestochastic difference equation (26) when we consider thelimiting case 120585 rarr 0

3 Time-Discrete Strongly Nonlinear StochasticProcesses on Discrete State Spaces

31 The Strongly Nonlinear Two-State Markov Chain ModelA fundamental strongly nonlinear stochastic process is thetwo-state nonlinear Markov chain process [17] For the sakeof convenience the states 119896 = 1 and 119896 = 2will be labeled withldquo119860rdquo and ldquo119861rdquo Transitions from119860 to119860119860 to 119861 119861 to119860 and 119861 to119861 occur with conditional probabilities 119879(119860 rarr 119860) 119879(119860 rarr

119861) 119879(119861 rarr 119860) and 119879(119861 rarr 119861) respectively Accordinglythe nonlinear Markov chain equation (7) reads

119901 (119860 119905 + 1) = 119879 (119860 997888rarr 119860 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119905)) 119901 (119861 119905)

119901 (119861 119905 + 1) = 119879 (119860 997888rarr 119861 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119861 119905 119901 (119905)) 119901 (119861 119905)

(36)

Taking the normalization condition 119901(119860) + 119901(119861) = 1 intoconsideration (36) can equivalently be expressed by

119901 (119860 119905 + 1)

= [119879 (119860 997888rarr 119860 119905 119901 (119860 119905)) minus 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))] 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))

(37)

which is a closed nonlinear equation in 119901(119860 119905) Model (37)involves two functions only the conditional probabilities119879(119860 rarr 119860) and 119879(119861 rarr 119860) which can be chosenindependently from each other and in the case of a nonlinearMarkov process depend on 119901(119860 119905) as indicated in (37)The remaining conditional probabilities can be calculatedfrom 119879(119860 rarr 119861) = 1 minus 119879(119860 rarr 119860) and 119879(119861 rarr

119861) = 1 minus 119879(119861 rarr 119860) Model (37) has been examined inphysics social sciences and psychology as will be discussednext

32 Chaos in Probability Time-Discrete Case FollowingFrank [18] let us put 119879(119860 rarr 119860) = 119879(119861 rarr 119860) = 119879

0 Then

(37) becomes

119901 (119860 119905 + 1) = 1198790(119905 119901 (119860 119905)) (38)

Although (38) describes a stochastic process formally (38)exhibits the structure of a nonlinear deterministic mapsubjected to the constraint that the function 119879

0must be in

the interval [0 1] at any time step 119905 and for any occupationprobability 119901(119860) Iterative maps in general are known toexhibit chaos [19] Therefore (38) suggests that time-discretestrongly nonlinear Markov processes exhibit chaos as wellSuch strongly nonlinear chaotic processes exhibit chaos in theevolution of the occupation probabilities and the conditionalprobabilities For example in Frank [18] the logistic function1198790(119911) = 120572 sdot 119911(1 minus 119911) [19 20] was considered for which (38)

reads

119901 (119860 119905 + 1) = 120572119901 (119860 119905) [1 minus 119901 (119860 119905)] (39)

For 120572 in [0 4] the function 1198790(sdot) is in the interval [0 1] such

that (39) maps the occupation probability 119901(119860) from theinterval [0 1] to the interval [0 1] as required For parameters120572 ge 120572crit asymp 3570 [20] the map exhibits chaotic solutionsThat is 119901(119860 119905) evolves in a chaotic fashion However asstated previously not only the occupation probability exhibitschaos but the conditional probabilities 119879(119860 rarr 119860) =

119879(119861 rarr 119860) = 1198790exhibit a chaotic behavior as well which

was demonstrated explicitly by numerical simulations [18]Chaos in strongly nonlinear time-discrete Markov processesis not limited to the two-state models In fact the two-statemodel can be generalized to an 119873-state model In this case(38) becomes [18]

119901 (119896 119905 + 1) = 1198790(119905 119901 (119905)) (40)

for 119896 = 1 119873 minus 1 The occupation probability 119901(119873 119905)

is computed from the normalization condition Conse-quently 119879

0depends only on the occupation probabilities

119901(1 119905) 119901(119873 minus 1 119905) For example the two-dimensionalchaotic Bakermap [20] was described bymeans of (40) using119873 = 3 [18]

33 Social Sciences andGroupDecisionMaking Time-DiscreteCase Group decision making has frequently been describedby nonlinear stochastic processes [21ndash24]The reason for thisis that under certain circumstances individuals are likely tobe affected by the opinion that is presented by a group asa whole rather than by the opinion of individuals This so-called group opinion in turn is produced by the opinions anddecisions of the individual group members Consequentlythe decision making process of an individual is affected bythe sample average obtained from the opinions of the othersIf the groups can be considered as being relatively large thesample averagemay be replaced by the ensemble averageThedecision making behavior of an individual is then consideredas a realization of a stochastic process that is affected by theensemble average of the process In the context of votingbehavior the two-state model introduced in Section 31 was

8 ISRNMathematical Physics

applied [17]Themembers of theUSHouse of Representativeshad the choice to vote for or against passing the so-calledbailout bill (US House of Representatives USA 2008) Thebill failed to pass in the September 29 vote In a second voteon October 3 the bill finally passedThat is a critical numberof ldquoNordquo voters changed their opinions between September 29and October 3 The states 119896 = 1 and 119896 = 2 were labeled byldquo119884rdquo and ldquo119873rdquo representing ldquoYesrdquo and ldquoNordquo voters Accordingto model (37) the conditional probabilities 119879(119873 rarr 119884) and119879(119884 rarr 119884) describing a change in the opinion from ldquoNordquo toldquoYesrdquo and a consistent ldquoYesrdquo opinion were considered Usingthe notation of (37) the voting model proposed in Frank [18]reads

119901 (119884 119905 + 1)

= [119879 (119884 997888rarr 119884 119905 119901 (119884 119905)) minus 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))] 119901 (119884 119905)

+ 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))

(41)

The data available in the literature suggested that 119879(119884 rarr

119884) = 1 For the119879(119873 rarr 119884) the function119879(119873 rarr 119884) = 119860minus119861sdot

119901(119884 119905) was used to describe the impact of the group opinionon the voting behavior This relationship assumes that thepressure to change from a ldquoNordquo voter to a ldquoYesrdquo voter was highimmediately subsequent to the September 29 votes when theprobability 119901(119884 119905) at 119905 = Sept 29 was not high enough tomakethe bill pass In this context it should be mentioned that thefailure of the bill to pass on September 29 triggered a dramaticbreakdown of the US stock market That is the members ofthe House of Representatives who had voted with ldquoNordquo werenot only subjected to political pressure but also experiencedessential pressure from the economy For 119879(119884 rarr 119884) = 1

and 119879(119873 rarr 119884) = 119860 minus 119861119901(119884 119905) the stationary occupationprobability 119901(119884 st) is given by 119901(119884 st) = 119860119861 such that (41)eventually reads

119901 (119884 119905 + 1) = 119901 (119884 119905) + 119861 [119901 (119884 st) minus 119901 (119884 119905)] [1 minus 119901 (119884 119905)]

(42)

The stationary value 119901(119884 st) was estimated from the dataA detailed model-based analysis of the data showed thatthe members of the two main parties the Democratic andRepublican members were differently affected by the grouppressure

34 Psychology and Human Memory A pioneering experi-ment in humanmemory research is the free recall experimentin which participants are asked to recall items of a particularcategory [25 26] For example they are asked to recall animalnames Typically participants are instructed to recall as manyitems as possible and to do so as fast as possible and withoutrepetitionThe total number of recalled items is by definitiona monotonically increasing function of time Strikinglyfree recall involves clusters in which item subcategories arerecalled more or less as a whole For example when recallinganimal names a participant may recall a number of insectnames in succession From a theoretical point of view at anygiven time point during the free recall process the ability

to recall items in clusters depends on the size of the poolof items available for recalling In other words if there isa large pool of items that have not yet been recalled thenthere is a high chance that a whole item subcategory canbe recalled In line with this argument a stochastic modelfor free recall dynamics was developed in which the recallprocess depends on the size of the pool of items availablefor recall [27] More precisely time was parceled in discretetime intervals in which recall attempts are executed The aimwas to develop a model for the probability that a single itemhas been or has not yet been recalled at a given time step 119905To this end the two-state model of Section 31 was appliedAccordingly the states 119896 = 1 and 119896 = 2 correspond tobeing recalled or being not yet been recalled and are labeledby ldquo119877rdquo and ldquo119873rdquo respectively The two relevant conditionalprobabilities are119879(119877 rarr 119877) and119879(119873 rarr 119877) By the design ofthe experiment we have119879(119877 rarr 119877) = 1 Moreover as arguedpreviously if the pool of items that have not yet recalled islarge that is if the probability 119901(119873) that an item has notyet been recalled is high then the conditional probability119879(119873 rarr 119877) should be high aswell Likewise if the probability119901(119877) that an item has been recalled is high then 119879(119873 rarr 119877)

should be relatively low Therefore it was suggested to put119879(119873 rarr 119877) = 119861 sdot (1 minus 119901(119877 119905)) with 119861 gt 0 This functionimplies that the occupation probability 119901(119877 119905) converges tothe stationary fixed 119901(119877 st) = 1 at which 119879(119873 rarr 119877) = 0The full model can be obtained by substituting 119879(119877 rarr 119877)

and 119879(119873 rarr 119877) into (37) and reads

119901 (119877 119905 + 1) = 119901 (119877 119905) + 119861[1 minus 119901 (119877 119905)]2

(43)

Note that from a mathematical point of view this modelis a special case of model (42) for voting behavior (hintput 119901(119884 st) = 1 in (42)) Since the model describes thesingle item recall probability the model must be scaled tothe number of items recalled by any given participant Tothis end a parameter 119862 describing the maximal number ofitems for recall was introduced The unknown parameters119861 and 119862 were estimated for free recall data collected in astudy by Rhodes and Turvey [28] and the sample meanvalues 119861 = 0009 (SD = 0005) and 119862 = 307 (SD = 138)were found for a sample of eight participants [27] Note that(43) predicts that the fixed point 119901(119877 st) = 1 will neverbe reached in a finite number of time steps Consequentlythe model predicts that the total number of recalled items119872 at the end of the experiment is smaller than the numberof available items for recall This prediction was confirmedby the data Moreover the model-based analysis revealed astatistically significant positive correlation between 119872 and119862 for the participant sample studied in Frank and Rhodes[27]

35 Biology and Evolutionary Population Dynamics A pri-mary concern of evolutionary population dynamics is theunderstanding of the origin of life Among the many modelsthat have been proposed in this context the studies byCrutchfield and Gornerup [29] and Gornerup and Crutch-field [30] are of interest for our review In these studiesa number of 119873 macromolecules with different functional

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Stochastic AnalysisInternational Journal of

Page 3: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 3

to stress the analogies between the classes listed in Table 1Mathematically speaking the probability density 119875 can bedefined by

119875 (119909) = lim119877rarrinfin

1

119877120575 (119883119903

1minus 1199091) sdot sdot sdot 120575 (119883

119903

119873minus 119909119873) (3)

In (3) the variable 119909 is the state vector with coordinates1199091 119909

119873 whereas the elements119883119903

1 119883

119903

119873are the elements

of the 119903th vector-valued realization 119883119903 of the process The

probability density satisfies the normalization condition

int

Ωtot

119875 (119909)

119873

prod

119896=1

119889119909119896= 1 Ωtot =

119873

prod

119896=1

Ω119896 (4)

where Ω119896are the domains of definitions mentioned previ-

ouslyLet us return to processes defined on a discrete state space

In addition we put119873 = 1 and assume thatΩtot = Ω1denotes

the real half-line from 0 to infinity Formally the probabilitydensity 119875(119909) assumes the form

119875 (119909) =

119873

sum

119896=1

119901 (119896) 120575 (119909 minus 119896) (5)

where 119901(119896) are the occupation probabilities of the states 119896defined in (1) In turn the occupation probabilities can becalculated from 119875(119909) like

119901 (119896) = int

119896+120576

119896minus120576

119875 (119909) 119889119909 (6)

where 120576 is a positive number smaller than 1 (eg 120576 = 01)In view of (6) the occupation probabilities 119901(119896)may be con-sidered as functions of an appropriately defined probabilitydensity119875 as stated in Section 1With these notations at handsome benchmark examples of strongly nonlinear stochasticprocesses can be given

22 Strongly Nonlinear Markov Processes Described by Non-linear Markov Chains In general the occupation proba-bilities 119901(119896) of time-discrete stochastic processes evolvingon discrete state spaces constitute a time-dependent vector119901(119905) with coefficients 119901(1 119905) 119901(119873 119905) For the importantspecial case of nonlinear Markov processes described bynonlinearMarkov chains the occupation probabilities satisfy[12]

119901 (119896 119905 + 1) =

119873

sum

119894=1

119879 (119894 997888rarr 119896 119905 119901 (119905)) 119901 (119894 119905) (7)

Here 119879 describes the transition probability matrix of theprocess The matrix is an119873 times119873matrix with coefficients119879(119894 rarr 119896) The coefficients 119879(119894 rarr 119896) describe theconditional probabilities that transitions from state 119894 to state119896 occur given that the process is in the state 119894The conditionalprobabilities 119879(119894 rarr 119896)may depend explicitly on time 119905 suchthat 119879(119894 rarr 119896 119905) A nonlinear Markov process is character-ized by the property that at least one conditional probability

depends at least on one expectation value computed from theprocess at time step 119905 or on one occupation probability119901(119896 119905)In this case we have 119879(119894 rarr 119896 119905 119901(119905)) as indicated in (7) Acomplete description of the stochastic process can be derivedfrom (7) as shown in Frank [12] In view of the definition ofa strongly nonlinear stochastic process given in Section 12that is based on realizations of stochastic processes it is worthwhile to consider an alternative definition of the process Tothis end the stochastic map 119891(sdot 120576) can be defined that mapsa state119883(119905) to a state119883(119905 + 1) like

119883 (119905 + 1) = 119891 (119883 (119905) 119905 119901 (119905) 120576 (119905)) (8)

The mapping 119891(sdot 120576) depends on a random variable 120576(119905) thatis uniformly distributed on the interval [0 1] at any timestep 119905 Across time 120576(119905) is statistically independent Thestochastic map 119891(sdot 120576) describes the jumps from119883(119905) to119883(119905+1) satisfying the transition probabilities 119879(119894 rarr 119896) In orderto derive an explicit definition of 119891(sdot 120576) we first note thatthe conditional probabilities 119879(119894 rarr 119896) are by definition realnumbers between 0 and 1 that add up to unity Next it ususeful to consider the cumulative conditional probabilities119878(119894 119896) defined for 119894 = 1 2 119873 and 119896 = 0 1 119873 by

119878 (119894 119896) =

119896

sum

119895=1

119879 (119894 997888rarr 119895 119905 119901 (119905)) (9)

for 119896 gt 0 For 119896 = 0 we put 119878(119894 0) = 0 For 119896 gt 0

the cumulativemeasures 119878(119894 119896) describe the probabilities thattransitions from state 119894 to one of the states 1 2 119896 occur attime 119905 given that the process has an occupation probabilityvector 119901 at time 119905 The cumulative measures represent aset of monotonically increasing points on the interval [0 1]such that the interval between two adjacent points 119878(119894 119896 minus

1) and 119878(119894 119896) equals 119879(119894 rarr 119896) Consequently the pointspectrum 119878(119894 119896) in combinationwith the randomvariable 120576(119905)can be used to construct realizations of the process underconsideration If 119883(119905) = 119894 holds and if 120576(119905) falls into theinterval [119878(119894 119896 minus 1) 119878(119894 119896)) then we put119883(119905+ 1) = 119896 If we doso then transitions from 119894 to 119896 happen with the probability119879(119894 rarr 119896) if the process is in state 119894 at 119905 and distributed like119901(119905) at 119905mdashas required A concise mathematical formulationof this idea can be obtained with the help of the indicatorfunction 119868(119894 rarr 119896 120576) defined by

119868 (119894 997888rarr 119896 119905 119901 120576) = 1 for 120576 isin [119878 (119894 119896 minus 1) 119878 (119894 119896))

0 otherwise(10)

Then the map 119891(sdot 120576) can be defined as follows

119891 (119883 (119905) = 119894 119905 119901 (119905) 120576 (119905)) =

119873

sum

119896=1

119896 sdot 119868 (119894 997888rarr 119896 119905 119901 (119905) 120576 (119905))

(11)

As indicated in (11) the mapping depends on the probabilityvector 119901(119905) because the indicator function 119868(sdot) depends on119901(119905) via the conditional probabilities 119878 and 119879 Thereforeeach realization 119883

119903

(119905) of the process defined by (8)ndash(11)depends on the occupation probabilities119901(119896 119905) of the process

4 ISRNMathematical Physics

The occupation probabilities in turn can be computed fromthe realizations see (1) Therefore the mathematical modelgiven by (1) and (8)ndash(11) provides a closed (and complete)description for the strongly nonlinear stochastic processdescribed by the nonlinear Markov chain (7)

23 Strongly Nonlinear Markov Processes Described by Sto-chastic Iterative Maps The realization of a time-discrete sto-chastic process on an119873-dimensional continuous state spaceis described by a time-dependent state vector119883(119905)with time-dependent components119883

1(119905) 119883

119873(119905) From (3) it follows

that at any time step 119905 the vector119883(119905) is distributed accordingto the time-dependent probability density

119875 (119909 119905) = lim119877rarrinfin

1

119877120575 (119883119903

1(119905) minus 119909

1) sdot sdot sdot 120575 (119883

119903

119873(119905) minus 119909

119873)

(12)

Strongly nonlinear Markov processes defined by stochasticiterative maps with nonparametric white noise satisfy equa-tions of the general form

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(13)

A simplified version of (13) was introduced earlier in Frank[13] In (13) the vector-valued variable ℎ(sdot) is called the driftfunction and describes the deterministic evolution of theprocess Second the term 119892(sdot) sdot 120576(119905) in (13) describes thevector-valued fluctuating force of the stochastic process Therandom vector 120576(119905) with coefficients 120576

1(119905) 120576

119873(119905) is an 119873-

dimensional normally distributed white noise process Thatis each coefficient 120576

119896(119905) is normally distributed at any time

step 119905 and statistically independent in time The randomcoefficients 120576

119896(119905) are statistically independent among each

otherThe variable 119892 is an119873 times119873matrix with coefficients119892119895119896 The coefficients 119892

119895119896with 119896 = 1 119873 for a fixed index

119895 weight the contributions that the random components1205761(119905) 120576

119873(119905) have on the evolution of the process in the

direction (or along the coordinate) 119909119895 In particular for

119892119895119896(sdot 119905) = 0 the random coefficient 120576

119896(119905) does not occur

in the evolution equation of 119883119895at time 119905 Therefore 119892 may

be considered as weight matrix Importantly the coefficients119892119895119896

measure the strength of the fluctuating force 119892(sdot) sdot 120576(119905)of the stochastic process described by (13) The reason forthis is that the random vector 120576(119905) is normalized at any timestep 119905 Most importantly in the case of a strongly nonlinearMarkov process the drift function ℎ(sdot) or the fluctuating force119892(sdot) sdot 120576(119905) or both depend on at least one expectation value oron the probability density 119875 as indicated in (13)The operator120579 maps the probability density 119875 to a real number or a set ofreal numbers This mapping may depend on the value of 119883For example the probability density may be evaluated at thevalue of the process like

120579119883(119905)

[119875 (119909 119905)] = 119875 (119909 119905)|119909=119883(119905)

(14)

In this context it should be noted that the formal depen-dencies ℎ(120579

119883[119875]) and 119892(120579

119883[119875]) include the dependencies on

expectation values (such as the mean) as special cases Forexample

120579119883(119905)

[119875 (119909 119905)] = int

Ωtot

119909119875 (119909 119905)

119873

prod

119896=1

119889119909119896= ⟨119883 (119905)⟩ (15)

where ⟨sdot⟩ is the operator symbol for an ensemble average suchthat ⟨119883⟩ denotes the mean of the random vector 119883 Notethat (12) and (13) provide a closed description of a stochasticprocess Moreover from the structure of (13) it followsimmediately that processes defined by (12) and (13) satisfythe definition of strongly nonlinear stochastic processes pre-sented in Section 12 An illustrative example of a stochasticiterative map (13) is the autoregressive model of order 1subjected to a so-called mean field force ℎMF proportional tothe mean ⟨119883(119905)⟩ of the process For 119873 = 1 the model reads[13]

119883 (119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(16)

with 119892 ge 0 where the parameters119860 and 119861 are scalarsWe willreturn to model (16) in Section 4

24 Strongly Nonlinear Markov Processes Described by Non-linearMaster Equations For time-continuous stochastic pro-cesses evolving on discrete state spaces the occupation prob-abilities 119901(119896) are functions of time 119905 where 119905 is a continuousvariable Likewise the probability vector 119901(119905) becomes atime-dependent variable with coefficients 119901(1 119905) 119901(119873 119905)

just as in Section 22 but with 119905 being a continuous ratherthan a discrete variable Time-continuous strongly nonlinearMarkov processes on discrete state spaces can be describedbymeans of nonlinear master equations of the form (see eg[14])

119901 (119896 119905) = 119901 (119896 1199051015840

)

+ int

119905

1199051015840

[

119873

sum

119894=1

119908 (119894 997888rarr 119896 119904 119901 (119904)) sdot 119901 (119894 119904)

minus119901 (119896 119904)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119904 119901 (119904))] 119889119904

(17)

for 119905 gt 1199051015840 and 119896 = 1 119873 with 119908(119896 rarr 119896) = 0 for all

states 119896 Here the variables119908(119894 rarr 119896) ge 0 are transition ratesthat describe the rate of transitions (number of transitions perunit time) from states 119894 to states 119896 We will use the convention119908(119896 rarr 119896) = 0 for all states 119896 Note that as opposed to theconditional probabilities 119879(sdot) occurring in a Markov chainthe rates 119908(sdot) of the master equation are not normalizedTransition rates may depend explicitly on time such that119908(119894 rarr 119896 119905) For strongly nonlinear stochastic processestransition rates do not only depend on the states 119894 and 119896

and on time but also depend on an expectation value or theoccupation probabilities 119901(119896 119905) Note that (17) is the integral

ISRNMathematical Physics 5

form of the master equation If the integral is evaluated for asmall time interval Δ119905 = 119905 minus 119905

1015840

rarr 0 then the integral versionexhibits the structure of a Markov chain (7) with

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905)) = Δ119905 sdot 119908 (119894 997888rarr 119896 119905 119901 (119905)) 119894 = 119896

119879 (119896 997888rarr 119896 119905 Δ119905 119901 (119905)) = 1 minus Δ119905

119873

sum

119895=1

119908 (119896 997888rarr 119895 119905 119901 (119905))

(18)

which can be written in a more concise form like

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905))

= Δ119905 119908 (119894 997888rarr 119896 119905 119901 (119905)) + 120575 (119894 119896)

sdot [

[

1 minus Δ119905

119873

sum

119895=1

119908 (119894 997888rarr 119895 119905 119901 (119905))]

]

(19)

Note again that we will use the convention that the diagonalelements 119908(119896 rarr 119896) of the transition rates equal zero whichimplies that for the diagonal elements 119879(119896 rarr 119896) only thesecond term in (19) matters as stated in (18) For the sake ofcompleteness note that the differential version of the masterequation (17) reads

119889

119889119905119901 (119896 119905) =

119873

sum

119894=1

119908 (119894 997888rarr 119896 119905 119901 (119905)) 119901 (119894 119905)

minus 119901 (119896 119905)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119905 119901 (119905))

(20)

Just as in Section 22 in order to describe the four classes ofstochastic processes listed in Table 1 on an equal footing weconsider an alternative description of the process defined by(20) Accordingly we introduce the flow functionΦ(119905 120576) thatmaps initial states119883(0) defined on the integer set 1 119873 tostates119883(119905)

119883 (119905) = Φ (119883 (0) 120576) (21)

The variable 120576 is a random variable that is uniformly dis-tributed on the interval [0 1] at every time point 119905 and isstatistically independently distributed across time In view ofthe fact that the time-discrete version of the master equationexhibits the structure of a Markov chain trajectories definedby the flow function can be regarded as time-continuouscounterparts to the trajectories of Markov chains as definedby (1) and (8)ndash(11)This analogy can be exploited to constructiteratively the flow function Φ in the limiting case Δ119905 rarr 0

like

Φ 119883 (119905 + Δ119905) = 119891 (119883 (119905) 119905 Δ119905 119901 (119905) 120576 (119905)) (22)

with

119891 (119883 (119905) = 119894 119905 Δ119905 119901 (119905) 120576 (119905))

=

119873

sum

119896=1

119896 sdot 119868 (119894 997888rarr 119896 119905 Δ119905 119901 (119905) 120576 (119905))

(23)

The iterative map 119891(sdot) involves the indicator function

119868 (119894 997888rarr 119896 120576 119905 Δ119905 119901) = 1 for 120576 isin [119878 (119894 119896 minus 1) 119878 (119894 119896))

0 otherwise(24)

and the cumulative transition probabilities

119878 (119894 119896) =

119896

sum

119895=1

119879 (119894 997888rarr 119895 119905 Δ119905 119901 (119905)) (25)

Equations (1) (18) and (22)ndash(25) provide a closed approxi-mate description of the trajectories of interest By definitionthe description becomes exact in the limiting case of infinitelysmall time intervals Δ119905 Note that the conditional (transition)probabilities 119879(sdot) defined by (18) are in the interval [0 1]provided that Δ119905 is chosen sufficiently small In the limitingcase Δ119905 rarr 0 this is always the case assuming that the rates119908(sdot) are bounded from above Note also that the probabilities119879(sdot) defined by (18) are normalized to 1 Furthermore from(18) and (22)ndash(25) it follows that 119891(sdot) = 119894 holds for Δ119905 = 0That is the iterativemap (22) describes transitions from states119883(119905) = 119894 to states119883(119905+Δ119905) in the small time intervalΔ119905 Mostimportantly according to (22) the evolution of realizationsdepends on the occupation probability vector 119901(119905) at everytime point 119905 Therefore processes defined by (1) (18) and(22)ndash(25) belong to the class of strongly nonlinear stochasticprocesses (see the definition in Section 12)

25 Strongly Nonlinear Markov Diffusion Processes Defined byStrongly Nonlinear Langevin Equations and Strongly Nonline-ar Fokker-Planck Equations Realizations of time-continuousstochastic processes evolving on119873-dimensional continuousstate spaces are described by time-dependent state vectors119883(119905) with time-dependent components 119883

1(119905) 119883

119873(119905) At

every time point 119905 the vector 119883(119905) is distributed accordingto the probability density 119875(119909 119905) defined by (12) where thevariable 119909 is the state vector with coordinates 119909

1 119909

119873

Strongly nonlinear Markov diffusion processes have realiza-tions (or trajectories) defined in the limiting case of infinitelysmall time intervals Δ119905 rarr 0 by strongly nonlinear stochasticdifference equations [9] of the form

119883 (119905 + Δ119905) = 119883 (119905) + Δ119905ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ radic2Δ119905119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(26)

Carrying out the limiting procedure (26) can be writtenin a more concise way in terms of a so-called Ito-Langevinequation

119889

119889119905119883 (119905) = ℎ (119883 (119905) 119905 120579

119883(119905)[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot Γ (119905)

(27)

Just as in the case of the stochastic iterative maps discussed inSection 23 (26) and (27) exhibit two different components

6 ISRNMathematical Physics

First the vector-valued function ℎ(sdot) occurring in (26) and(27) is the drift function introduced in Section 23 thataccounts for the deterministic evolution of a process Secondthe expressions 119892 sdot 120576 and 119892 sdot Γ respectively describe avector-valued fluctuating force The variable 120576(119905) is the 119873-dimensional random variable introduced in Section 23 Thefunction Γ(119905) in the Ito-Langevin equation (27) is the coun-terpart to the random vector 120576(119905) in (26) and corresponds to aso-called vector-valued Langevin force [4 6] More preciselyeach component Γ

119896(119905) is a Langevin force We will consider

Langevin forces normalized with respect to a magnitude of 2like

lim119877rarrinfin

1

119877Γ119903

119895(119905) Γ119903

119896(119904) = 2120575 (119895 119896) 120575 (119905 minus 119904) (28)

where Γ119903

119896(119905) are realizations of the 119896th component The

variable 119892 is the 119873 times 119873 weight matrix introduced inSection 23

The stochastic process defined by the Ito-Langevin equa-tion (27) can equivalently be expressed bymeans of a stronglynonlinear Fokker-Planck equation

120597

120597119905119875 (119909 119905) = minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119875 (119909 119905)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119875 (119909 119905)]

(29)

The evolution equation for 119875(119909 119905) is nonlinear for 119875(119909 119905)In contrast the corresponding evolution equation for theconditional probability density 119879(119909 119905 | 119910 119904) with 119905 gt 119904 islinear with respect to 119879(119909 119905 | 119910 119904) and reads

120597

120597119905119879 (119909 119905 | 119910 119904)

= minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119879 (119909 119905 | 119910 119904)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119879 (119909 119905 | 119910 119904)]

(30)

For details see Frank [9] A complete description of thestochastic process can be obtained either from the realiza-tions defined by (26) and (27) or from the evolution equations(29) and (30) for 119875(119909 119905) and 119879(119909 119905 | 119910 119904) The expression

119863119894119895(119909 119905 120579

119909[119875 (sdot 119905)]) =

119873

sum

119896=1

119892119894119896(sdot) 119892119895119896(sdot) (31)

occurring in (29) and (30) is the diffusion coefficient of theprocess

A fundamental example of a strongly nonlinear Markovdiffusion process is a generalized Ornstein-Uhlenbeck pro-cess involving a mean field force ℎMF proportional to the

mean ⟨119883(119905)⟩ of the process [9 15 16] For 119873 = 1 thestochastic difference equation reads

119883 (119905 + Δ119905) = 119883 (119905) minus Δ119905 [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + radic2Δ119905119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(32)

with 119892 ge 0 Typically the case119860 119861 gt 0 is considered For (32)the strongly nonlinear Langevin equation reads

119889

119889119905119883 (119905) = minus [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + 119892Γ (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(33)

Finally the corresponding strongly nonlinear Fokker-Planckequation reads

120597

120597119905119875 (119909 119905) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119875 (119909 119905)]

+ 1198631205972

1205971199092119875 (119909 119905)

120597

120597119905119879 (119909 119905 | 119910 119904) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119879 (119909 119905 | 119910 119904)]

+ 1198631205972

1205971199092119879 (119909 119905 | 119910 119904)

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

119863 = 1198922

(34)

The generalized Ornstein-Uhlenbeck process in its form asstochastic difference equation (32) corresponds to a specialcase of the generalized autoregressive model of order 1defined by (16) Model (34) has been studied in the contextof the Shimizu-Yamada model We will return to this issuelater

26 On a General Stochastic Evolution Equation for StronglyNonlinear Markov Processes In the previous sections weconsidered some benchmark examples of strongly nonlinearstochastic processes belonging to the four classes of processesdefined in Table 1 All processes considered in Sections 22to 25 satisfy the Markov property A more general formaldescription of strongly nonlinear Markov processes thatholds for all four classes reads

119883 (119905 + 120585) = 119891 (119883 (119905) 119905 120585 120579119883(119905)

[119875 (119909 119905)] 120576 (119905)) (35)

The additional variable 120585 defines the characteristic jumpinginterval of the process under consideration and can be used todistinguish between time-discrete and time-continuous pro-cesses For 120585 = 1 we are dealing with time-discrete stochastic

ISRNMathematical Physics 7

processes In the case of processes defined on discrete statespaces 119875(119909 119905) is related to the occupation probability 119901(119896 119905)by means of (5) and (6) and (35) reduces to the special caseof (8) of the Markov chain model For processes defined oncontinuous state spaces and 120585 = 1 the general form (35)becomes (13) in the case of stochastic processes driven bynonparametric white noise In the limiting case 120585 rarr 0 weare dealing with time-continuous stochastic processes In thiscase the function 119891(sdot) possesses the property 119891(sdot) = 119883(119905)

for 120585 = 0 For processes that evolve on discrete state spacesand satisfy a nonlinear master equation it follows that (35)becomes the iterative map (22) provided that the argument119875(119909 119905) in 119891(sdot) is expressed in terms of the occupationprobability 119901(119896 119905) see (6) In contrast for processes evolvingon continuous state spaces and satisfying strongly nonlinearIto-Langevin equations the general form (35) reduces to thestochastic difference equation (26) when we consider thelimiting case 120585 rarr 0

3 Time-Discrete Strongly Nonlinear StochasticProcesses on Discrete State Spaces

31 The Strongly Nonlinear Two-State Markov Chain ModelA fundamental strongly nonlinear stochastic process is thetwo-state nonlinear Markov chain process [17] For the sakeof convenience the states 119896 = 1 and 119896 = 2will be labeled withldquo119860rdquo and ldquo119861rdquo Transitions from119860 to119860119860 to 119861 119861 to119860 and 119861 to119861 occur with conditional probabilities 119879(119860 rarr 119860) 119879(119860 rarr

119861) 119879(119861 rarr 119860) and 119879(119861 rarr 119861) respectively Accordinglythe nonlinear Markov chain equation (7) reads

119901 (119860 119905 + 1) = 119879 (119860 997888rarr 119860 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119905)) 119901 (119861 119905)

119901 (119861 119905 + 1) = 119879 (119860 997888rarr 119861 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119861 119905 119901 (119905)) 119901 (119861 119905)

(36)

Taking the normalization condition 119901(119860) + 119901(119861) = 1 intoconsideration (36) can equivalently be expressed by

119901 (119860 119905 + 1)

= [119879 (119860 997888rarr 119860 119905 119901 (119860 119905)) minus 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))] 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))

(37)

which is a closed nonlinear equation in 119901(119860 119905) Model (37)involves two functions only the conditional probabilities119879(119860 rarr 119860) and 119879(119861 rarr 119860) which can be chosenindependently from each other and in the case of a nonlinearMarkov process depend on 119901(119860 119905) as indicated in (37)The remaining conditional probabilities can be calculatedfrom 119879(119860 rarr 119861) = 1 minus 119879(119860 rarr 119860) and 119879(119861 rarr

119861) = 1 minus 119879(119861 rarr 119860) Model (37) has been examined inphysics social sciences and psychology as will be discussednext

32 Chaos in Probability Time-Discrete Case FollowingFrank [18] let us put 119879(119860 rarr 119860) = 119879(119861 rarr 119860) = 119879

0 Then

(37) becomes

119901 (119860 119905 + 1) = 1198790(119905 119901 (119860 119905)) (38)

Although (38) describes a stochastic process formally (38)exhibits the structure of a nonlinear deterministic mapsubjected to the constraint that the function 119879

0must be in

the interval [0 1] at any time step 119905 and for any occupationprobability 119901(119860) Iterative maps in general are known toexhibit chaos [19] Therefore (38) suggests that time-discretestrongly nonlinear Markov processes exhibit chaos as wellSuch strongly nonlinear chaotic processes exhibit chaos in theevolution of the occupation probabilities and the conditionalprobabilities For example in Frank [18] the logistic function1198790(119911) = 120572 sdot 119911(1 minus 119911) [19 20] was considered for which (38)

reads

119901 (119860 119905 + 1) = 120572119901 (119860 119905) [1 minus 119901 (119860 119905)] (39)

For 120572 in [0 4] the function 1198790(sdot) is in the interval [0 1] such

that (39) maps the occupation probability 119901(119860) from theinterval [0 1] to the interval [0 1] as required For parameters120572 ge 120572crit asymp 3570 [20] the map exhibits chaotic solutionsThat is 119901(119860 119905) evolves in a chaotic fashion However asstated previously not only the occupation probability exhibitschaos but the conditional probabilities 119879(119860 rarr 119860) =

119879(119861 rarr 119860) = 1198790exhibit a chaotic behavior as well which

was demonstrated explicitly by numerical simulations [18]Chaos in strongly nonlinear time-discrete Markov processesis not limited to the two-state models In fact the two-statemodel can be generalized to an 119873-state model In this case(38) becomes [18]

119901 (119896 119905 + 1) = 1198790(119905 119901 (119905)) (40)

for 119896 = 1 119873 minus 1 The occupation probability 119901(119873 119905)

is computed from the normalization condition Conse-quently 119879

0depends only on the occupation probabilities

119901(1 119905) 119901(119873 minus 1 119905) For example the two-dimensionalchaotic Bakermap [20] was described bymeans of (40) using119873 = 3 [18]

33 Social Sciences andGroupDecisionMaking Time-DiscreteCase Group decision making has frequently been describedby nonlinear stochastic processes [21ndash24]The reason for thisis that under certain circumstances individuals are likely tobe affected by the opinion that is presented by a group asa whole rather than by the opinion of individuals This so-called group opinion in turn is produced by the opinions anddecisions of the individual group members Consequentlythe decision making process of an individual is affected bythe sample average obtained from the opinions of the othersIf the groups can be considered as being relatively large thesample averagemay be replaced by the ensemble averageThedecision making behavior of an individual is then consideredas a realization of a stochastic process that is affected by theensemble average of the process In the context of votingbehavior the two-state model introduced in Section 31 was

8 ISRNMathematical Physics

applied [17]Themembers of theUSHouse of Representativeshad the choice to vote for or against passing the so-calledbailout bill (US House of Representatives USA 2008) Thebill failed to pass in the September 29 vote In a second voteon October 3 the bill finally passedThat is a critical numberof ldquoNordquo voters changed their opinions between September 29and October 3 The states 119896 = 1 and 119896 = 2 were labeled byldquo119884rdquo and ldquo119873rdquo representing ldquoYesrdquo and ldquoNordquo voters Accordingto model (37) the conditional probabilities 119879(119873 rarr 119884) and119879(119884 rarr 119884) describing a change in the opinion from ldquoNordquo toldquoYesrdquo and a consistent ldquoYesrdquo opinion were considered Usingthe notation of (37) the voting model proposed in Frank [18]reads

119901 (119884 119905 + 1)

= [119879 (119884 997888rarr 119884 119905 119901 (119884 119905)) minus 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))] 119901 (119884 119905)

+ 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))

(41)

The data available in the literature suggested that 119879(119884 rarr

119884) = 1 For the119879(119873 rarr 119884) the function119879(119873 rarr 119884) = 119860minus119861sdot

119901(119884 119905) was used to describe the impact of the group opinionon the voting behavior This relationship assumes that thepressure to change from a ldquoNordquo voter to a ldquoYesrdquo voter was highimmediately subsequent to the September 29 votes when theprobability 119901(119884 119905) at 119905 = Sept 29 was not high enough tomakethe bill pass In this context it should be mentioned that thefailure of the bill to pass on September 29 triggered a dramaticbreakdown of the US stock market That is the members ofthe House of Representatives who had voted with ldquoNordquo werenot only subjected to political pressure but also experiencedessential pressure from the economy For 119879(119884 rarr 119884) = 1

and 119879(119873 rarr 119884) = 119860 minus 119861119901(119884 119905) the stationary occupationprobability 119901(119884 st) is given by 119901(119884 st) = 119860119861 such that (41)eventually reads

119901 (119884 119905 + 1) = 119901 (119884 119905) + 119861 [119901 (119884 st) minus 119901 (119884 119905)] [1 minus 119901 (119884 119905)]

(42)

The stationary value 119901(119884 st) was estimated from the dataA detailed model-based analysis of the data showed thatthe members of the two main parties the Democratic andRepublican members were differently affected by the grouppressure

34 Psychology and Human Memory A pioneering experi-ment in humanmemory research is the free recall experimentin which participants are asked to recall items of a particularcategory [25 26] For example they are asked to recall animalnames Typically participants are instructed to recall as manyitems as possible and to do so as fast as possible and withoutrepetitionThe total number of recalled items is by definitiona monotonically increasing function of time Strikinglyfree recall involves clusters in which item subcategories arerecalled more or less as a whole For example when recallinganimal names a participant may recall a number of insectnames in succession From a theoretical point of view at anygiven time point during the free recall process the ability

to recall items in clusters depends on the size of the poolof items available for recalling In other words if there isa large pool of items that have not yet been recalled thenthere is a high chance that a whole item subcategory canbe recalled In line with this argument a stochastic modelfor free recall dynamics was developed in which the recallprocess depends on the size of the pool of items availablefor recall [27] More precisely time was parceled in discretetime intervals in which recall attempts are executed The aimwas to develop a model for the probability that a single itemhas been or has not yet been recalled at a given time step 119905To this end the two-state model of Section 31 was appliedAccordingly the states 119896 = 1 and 119896 = 2 correspond tobeing recalled or being not yet been recalled and are labeledby ldquo119877rdquo and ldquo119873rdquo respectively The two relevant conditionalprobabilities are119879(119877 rarr 119877) and119879(119873 rarr 119877) By the design ofthe experiment we have119879(119877 rarr 119877) = 1 Moreover as arguedpreviously if the pool of items that have not yet recalled islarge that is if the probability 119901(119873) that an item has notyet been recalled is high then the conditional probability119879(119873 rarr 119877) should be high aswell Likewise if the probability119901(119877) that an item has been recalled is high then 119879(119873 rarr 119877)

should be relatively low Therefore it was suggested to put119879(119873 rarr 119877) = 119861 sdot (1 minus 119901(119877 119905)) with 119861 gt 0 This functionimplies that the occupation probability 119901(119877 119905) converges tothe stationary fixed 119901(119877 st) = 1 at which 119879(119873 rarr 119877) = 0The full model can be obtained by substituting 119879(119877 rarr 119877)

and 119879(119873 rarr 119877) into (37) and reads

119901 (119877 119905 + 1) = 119901 (119877 119905) + 119861[1 minus 119901 (119877 119905)]2

(43)

Note that from a mathematical point of view this modelis a special case of model (42) for voting behavior (hintput 119901(119884 st) = 1 in (42)) Since the model describes thesingle item recall probability the model must be scaled tothe number of items recalled by any given participant Tothis end a parameter 119862 describing the maximal number ofitems for recall was introduced The unknown parameters119861 and 119862 were estimated for free recall data collected in astudy by Rhodes and Turvey [28] and the sample meanvalues 119861 = 0009 (SD = 0005) and 119862 = 307 (SD = 138)were found for a sample of eight participants [27] Note that(43) predicts that the fixed point 119901(119877 st) = 1 will neverbe reached in a finite number of time steps Consequentlythe model predicts that the total number of recalled items119872 at the end of the experiment is smaller than the numberof available items for recall This prediction was confirmedby the data Moreover the model-based analysis revealed astatistically significant positive correlation between 119872 and119862 for the participant sample studied in Frank and Rhodes[27]

35 Biology and Evolutionary Population Dynamics A pri-mary concern of evolutionary population dynamics is theunderstanding of the origin of life Among the many modelsthat have been proposed in this context the studies byCrutchfield and Gornerup [29] and Gornerup and Crutch-field [30] are of interest for our review In these studiesa number of 119873 macromolecules with different functional

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

4 ISRNMathematical Physics

The occupation probabilities in turn can be computed fromthe realizations see (1) Therefore the mathematical modelgiven by (1) and (8)ndash(11) provides a closed (and complete)description for the strongly nonlinear stochastic processdescribed by the nonlinear Markov chain (7)

23 Strongly Nonlinear Markov Processes Described by Sto-chastic Iterative Maps The realization of a time-discrete sto-chastic process on an119873-dimensional continuous state spaceis described by a time-dependent state vector119883(119905)with time-dependent components119883

1(119905) 119883

119873(119905) From (3) it follows

that at any time step 119905 the vector119883(119905) is distributed accordingto the time-dependent probability density

119875 (119909 119905) = lim119877rarrinfin

1

119877120575 (119883119903

1(119905) minus 119909

1) sdot sdot sdot 120575 (119883

119903

119873(119905) minus 119909

119873)

(12)

Strongly nonlinear Markov processes defined by stochasticiterative maps with nonparametric white noise satisfy equa-tions of the general form

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(13)

A simplified version of (13) was introduced earlier in Frank[13] In (13) the vector-valued variable ℎ(sdot) is called the driftfunction and describes the deterministic evolution of theprocess Second the term 119892(sdot) sdot 120576(119905) in (13) describes thevector-valued fluctuating force of the stochastic process Therandom vector 120576(119905) with coefficients 120576

1(119905) 120576

119873(119905) is an 119873-

dimensional normally distributed white noise process Thatis each coefficient 120576

119896(119905) is normally distributed at any time

step 119905 and statistically independent in time The randomcoefficients 120576

119896(119905) are statistically independent among each

otherThe variable 119892 is an119873 times119873matrix with coefficients119892119895119896 The coefficients 119892

119895119896with 119896 = 1 119873 for a fixed index

119895 weight the contributions that the random components1205761(119905) 120576

119873(119905) have on the evolution of the process in the

direction (or along the coordinate) 119909119895 In particular for

119892119895119896(sdot 119905) = 0 the random coefficient 120576

119896(119905) does not occur

in the evolution equation of 119883119895at time 119905 Therefore 119892 may

be considered as weight matrix Importantly the coefficients119892119895119896

measure the strength of the fluctuating force 119892(sdot) sdot 120576(119905)of the stochastic process described by (13) The reason forthis is that the random vector 120576(119905) is normalized at any timestep 119905 Most importantly in the case of a strongly nonlinearMarkov process the drift function ℎ(sdot) or the fluctuating force119892(sdot) sdot 120576(119905) or both depend on at least one expectation value oron the probability density 119875 as indicated in (13)The operator120579 maps the probability density 119875 to a real number or a set ofreal numbers This mapping may depend on the value of 119883For example the probability density may be evaluated at thevalue of the process like

120579119883(119905)

[119875 (119909 119905)] = 119875 (119909 119905)|119909=119883(119905)

(14)

In this context it should be noted that the formal depen-dencies ℎ(120579

119883[119875]) and 119892(120579

119883[119875]) include the dependencies on

expectation values (such as the mean) as special cases Forexample

120579119883(119905)

[119875 (119909 119905)] = int

Ωtot

119909119875 (119909 119905)

119873

prod

119896=1

119889119909119896= ⟨119883 (119905)⟩ (15)

where ⟨sdot⟩ is the operator symbol for an ensemble average suchthat ⟨119883⟩ denotes the mean of the random vector 119883 Notethat (12) and (13) provide a closed description of a stochasticprocess Moreover from the structure of (13) it followsimmediately that processes defined by (12) and (13) satisfythe definition of strongly nonlinear stochastic processes pre-sented in Section 12 An illustrative example of a stochasticiterative map (13) is the autoregressive model of order 1subjected to a so-called mean field force ℎMF proportional tothe mean ⟨119883(119905)⟩ of the process For 119873 = 1 the model reads[13]

119883 (119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(16)

with 119892 ge 0 where the parameters119860 and 119861 are scalarsWe willreturn to model (16) in Section 4

24 Strongly Nonlinear Markov Processes Described by Non-linearMaster Equations For time-continuous stochastic pro-cesses evolving on discrete state spaces the occupation prob-abilities 119901(119896) are functions of time 119905 where 119905 is a continuousvariable Likewise the probability vector 119901(119905) becomes atime-dependent variable with coefficients 119901(1 119905) 119901(119873 119905)

just as in Section 22 but with 119905 being a continuous ratherthan a discrete variable Time-continuous strongly nonlinearMarkov processes on discrete state spaces can be describedbymeans of nonlinear master equations of the form (see eg[14])

119901 (119896 119905) = 119901 (119896 1199051015840

)

+ int

119905

1199051015840

[

119873

sum

119894=1

119908 (119894 997888rarr 119896 119904 119901 (119904)) sdot 119901 (119894 119904)

minus119901 (119896 119904)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119904 119901 (119904))] 119889119904

(17)

for 119905 gt 1199051015840 and 119896 = 1 119873 with 119908(119896 rarr 119896) = 0 for all

states 119896 Here the variables119908(119894 rarr 119896) ge 0 are transition ratesthat describe the rate of transitions (number of transitions perunit time) from states 119894 to states 119896 We will use the convention119908(119896 rarr 119896) = 0 for all states 119896 Note that as opposed to theconditional probabilities 119879(sdot) occurring in a Markov chainthe rates 119908(sdot) of the master equation are not normalizedTransition rates may depend explicitly on time such that119908(119894 rarr 119896 119905) For strongly nonlinear stochastic processestransition rates do not only depend on the states 119894 and 119896

and on time but also depend on an expectation value or theoccupation probabilities 119901(119896 119905) Note that (17) is the integral

ISRNMathematical Physics 5

form of the master equation If the integral is evaluated for asmall time interval Δ119905 = 119905 minus 119905

1015840

rarr 0 then the integral versionexhibits the structure of a Markov chain (7) with

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905)) = Δ119905 sdot 119908 (119894 997888rarr 119896 119905 119901 (119905)) 119894 = 119896

119879 (119896 997888rarr 119896 119905 Δ119905 119901 (119905)) = 1 minus Δ119905

119873

sum

119895=1

119908 (119896 997888rarr 119895 119905 119901 (119905))

(18)

which can be written in a more concise form like

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905))

= Δ119905 119908 (119894 997888rarr 119896 119905 119901 (119905)) + 120575 (119894 119896)

sdot [

[

1 minus Δ119905

119873

sum

119895=1

119908 (119894 997888rarr 119895 119905 119901 (119905))]

]

(19)

Note again that we will use the convention that the diagonalelements 119908(119896 rarr 119896) of the transition rates equal zero whichimplies that for the diagonal elements 119879(119896 rarr 119896) only thesecond term in (19) matters as stated in (18) For the sake ofcompleteness note that the differential version of the masterequation (17) reads

119889

119889119905119901 (119896 119905) =

119873

sum

119894=1

119908 (119894 997888rarr 119896 119905 119901 (119905)) 119901 (119894 119905)

minus 119901 (119896 119905)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119905 119901 (119905))

(20)

Just as in Section 22 in order to describe the four classes ofstochastic processes listed in Table 1 on an equal footing weconsider an alternative description of the process defined by(20) Accordingly we introduce the flow functionΦ(119905 120576) thatmaps initial states119883(0) defined on the integer set 1 119873 tostates119883(119905)

119883 (119905) = Φ (119883 (0) 120576) (21)

The variable 120576 is a random variable that is uniformly dis-tributed on the interval [0 1] at every time point 119905 and isstatistically independently distributed across time In view ofthe fact that the time-discrete version of the master equationexhibits the structure of a Markov chain trajectories definedby the flow function can be regarded as time-continuouscounterparts to the trajectories of Markov chains as definedby (1) and (8)ndash(11)This analogy can be exploited to constructiteratively the flow function Φ in the limiting case Δ119905 rarr 0

like

Φ 119883 (119905 + Δ119905) = 119891 (119883 (119905) 119905 Δ119905 119901 (119905) 120576 (119905)) (22)

with

119891 (119883 (119905) = 119894 119905 Δ119905 119901 (119905) 120576 (119905))

=

119873

sum

119896=1

119896 sdot 119868 (119894 997888rarr 119896 119905 Δ119905 119901 (119905) 120576 (119905))

(23)

The iterative map 119891(sdot) involves the indicator function

119868 (119894 997888rarr 119896 120576 119905 Δ119905 119901) = 1 for 120576 isin [119878 (119894 119896 minus 1) 119878 (119894 119896))

0 otherwise(24)

and the cumulative transition probabilities

119878 (119894 119896) =

119896

sum

119895=1

119879 (119894 997888rarr 119895 119905 Δ119905 119901 (119905)) (25)

Equations (1) (18) and (22)ndash(25) provide a closed approxi-mate description of the trajectories of interest By definitionthe description becomes exact in the limiting case of infinitelysmall time intervals Δ119905 Note that the conditional (transition)probabilities 119879(sdot) defined by (18) are in the interval [0 1]provided that Δ119905 is chosen sufficiently small In the limitingcase Δ119905 rarr 0 this is always the case assuming that the rates119908(sdot) are bounded from above Note also that the probabilities119879(sdot) defined by (18) are normalized to 1 Furthermore from(18) and (22)ndash(25) it follows that 119891(sdot) = 119894 holds for Δ119905 = 0That is the iterativemap (22) describes transitions from states119883(119905) = 119894 to states119883(119905+Δ119905) in the small time intervalΔ119905 Mostimportantly according to (22) the evolution of realizationsdepends on the occupation probability vector 119901(119905) at everytime point 119905 Therefore processes defined by (1) (18) and(22)ndash(25) belong to the class of strongly nonlinear stochasticprocesses (see the definition in Section 12)

25 Strongly Nonlinear Markov Diffusion Processes Defined byStrongly Nonlinear Langevin Equations and Strongly Nonline-ar Fokker-Planck Equations Realizations of time-continuousstochastic processes evolving on119873-dimensional continuousstate spaces are described by time-dependent state vectors119883(119905) with time-dependent components 119883

1(119905) 119883

119873(119905) At

every time point 119905 the vector 119883(119905) is distributed accordingto the probability density 119875(119909 119905) defined by (12) where thevariable 119909 is the state vector with coordinates 119909

1 119909

119873

Strongly nonlinear Markov diffusion processes have realiza-tions (or trajectories) defined in the limiting case of infinitelysmall time intervals Δ119905 rarr 0 by strongly nonlinear stochasticdifference equations [9] of the form

119883 (119905 + Δ119905) = 119883 (119905) + Δ119905ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ radic2Δ119905119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(26)

Carrying out the limiting procedure (26) can be writtenin a more concise way in terms of a so-called Ito-Langevinequation

119889

119889119905119883 (119905) = ℎ (119883 (119905) 119905 120579

119883(119905)[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot Γ (119905)

(27)

Just as in the case of the stochastic iterative maps discussed inSection 23 (26) and (27) exhibit two different components

6 ISRNMathematical Physics

First the vector-valued function ℎ(sdot) occurring in (26) and(27) is the drift function introduced in Section 23 thataccounts for the deterministic evolution of a process Secondthe expressions 119892 sdot 120576 and 119892 sdot Γ respectively describe avector-valued fluctuating force The variable 120576(119905) is the 119873-dimensional random variable introduced in Section 23 Thefunction Γ(119905) in the Ito-Langevin equation (27) is the coun-terpart to the random vector 120576(119905) in (26) and corresponds to aso-called vector-valued Langevin force [4 6] More preciselyeach component Γ

119896(119905) is a Langevin force We will consider

Langevin forces normalized with respect to a magnitude of 2like

lim119877rarrinfin

1

119877Γ119903

119895(119905) Γ119903

119896(119904) = 2120575 (119895 119896) 120575 (119905 minus 119904) (28)

where Γ119903

119896(119905) are realizations of the 119896th component The

variable 119892 is the 119873 times 119873 weight matrix introduced inSection 23

The stochastic process defined by the Ito-Langevin equa-tion (27) can equivalently be expressed bymeans of a stronglynonlinear Fokker-Planck equation

120597

120597119905119875 (119909 119905) = minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119875 (119909 119905)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119875 (119909 119905)]

(29)

The evolution equation for 119875(119909 119905) is nonlinear for 119875(119909 119905)In contrast the corresponding evolution equation for theconditional probability density 119879(119909 119905 | 119910 119904) with 119905 gt 119904 islinear with respect to 119879(119909 119905 | 119910 119904) and reads

120597

120597119905119879 (119909 119905 | 119910 119904)

= minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119879 (119909 119905 | 119910 119904)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119879 (119909 119905 | 119910 119904)]

(30)

For details see Frank [9] A complete description of thestochastic process can be obtained either from the realiza-tions defined by (26) and (27) or from the evolution equations(29) and (30) for 119875(119909 119905) and 119879(119909 119905 | 119910 119904) The expression

119863119894119895(119909 119905 120579

119909[119875 (sdot 119905)]) =

119873

sum

119896=1

119892119894119896(sdot) 119892119895119896(sdot) (31)

occurring in (29) and (30) is the diffusion coefficient of theprocess

A fundamental example of a strongly nonlinear Markovdiffusion process is a generalized Ornstein-Uhlenbeck pro-cess involving a mean field force ℎMF proportional to the

mean ⟨119883(119905)⟩ of the process [9 15 16] For 119873 = 1 thestochastic difference equation reads

119883 (119905 + Δ119905) = 119883 (119905) minus Δ119905 [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + radic2Δ119905119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(32)

with 119892 ge 0 Typically the case119860 119861 gt 0 is considered For (32)the strongly nonlinear Langevin equation reads

119889

119889119905119883 (119905) = minus [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + 119892Γ (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(33)

Finally the corresponding strongly nonlinear Fokker-Planckequation reads

120597

120597119905119875 (119909 119905) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119875 (119909 119905)]

+ 1198631205972

1205971199092119875 (119909 119905)

120597

120597119905119879 (119909 119905 | 119910 119904) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119879 (119909 119905 | 119910 119904)]

+ 1198631205972

1205971199092119879 (119909 119905 | 119910 119904)

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

119863 = 1198922

(34)

The generalized Ornstein-Uhlenbeck process in its form asstochastic difference equation (32) corresponds to a specialcase of the generalized autoregressive model of order 1defined by (16) Model (34) has been studied in the contextof the Shimizu-Yamada model We will return to this issuelater

26 On a General Stochastic Evolution Equation for StronglyNonlinear Markov Processes In the previous sections weconsidered some benchmark examples of strongly nonlinearstochastic processes belonging to the four classes of processesdefined in Table 1 All processes considered in Sections 22to 25 satisfy the Markov property A more general formaldescription of strongly nonlinear Markov processes thatholds for all four classes reads

119883 (119905 + 120585) = 119891 (119883 (119905) 119905 120585 120579119883(119905)

[119875 (119909 119905)] 120576 (119905)) (35)

The additional variable 120585 defines the characteristic jumpinginterval of the process under consideration and can be used todistinguish between time-discrete and time-continuous pro-cesses For 120585 = 1 we are dealing with time-discrete stochastic

ISRNMathematical Physics 7

processes In the case of processes defined on discrete statespaces 119875(119909 119905) is related to the occupation probability 119901(119896 119905)by means of (5) and (6) and (35) reduces to the special caseof (8) of the Markov chain model For processes defined oncontinuous state spaces and 120585 = 1 the general form (35)becomes (13) in the case of stochastic processes driven bynonparametric white noise In the limiting case 120585 rarr 0 weare dealing with time-continuous stochastic processes In thiscase the function 119891(sdot) possesses the property 119891(sdot) = 119883(119905)

for 120585 = 0 For processes that evolve on discrete state spacesand satisfy a nonlinear master equation it follows that (35)becomes the iterative map (22) provided that the argument119875(119909 119905) in 119891(sdot) is expressed in terms of the occupationprobability 119901(119896 119905) see (6) In contrast for processes evolvingon continuous state spaces and satisfying strongly nonlinearIto-Langevin equations the general form (35) reduces to thestochastic difference equation (26) when we consider thelimiting case 120585 rarr 0

3 Time-Discrete Strongly Nonlinear StochasticProcesses on Discrete State Spaces

31 The Strongly Nonlinear Two-State Markov Chain ModelA fundamental strongly nonlinear stochastic process is thetwo-state nonlinear Markov chain process [17] For the sakeof convenience the states 119896 = 1 and 119896 = 2will be labeled withldquo119860rdquo and ldquo119861rdquo Transitions from119860 to119860119860 to 119861 119861 to119860 and 119861 to119861 occur with conditional probabilities 119879(119860 rarr 119860) 119879(119860 rarr

119861) 119879(119861 rarr 119860) and 119879(119861 rarr 119861) respectively Accordinglythe nonlinear Markov chain equation (7) reads

119901 (119860 119905 + 1) = 119879 (119860 997888rarr 119860 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119905)) 119901 (119861 119905)

119901 (119861 119905 + 1) = 119879 (119860 997888rarr 119861 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119861 119905 119901 (119905)) 119901 (119861 119905)

(36)

Taking the normalization condition 119901(119860) + 119901(119861) = 1 intoconsideration (36) can equivalently be expressed by

119901 (119860 119905 + 1)

= [119879 (119860 997888rarr 119860 119905 119901 (119860 119905)) minus 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))] 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))

(37)

which is a closed nonlinear equation in 119901(119860 119905) Model (37)involves two functions only the conditional probabilities119879(119860 rarr 119860) and 119879(119861 rarr 119860) which can be chosenindependently from each other and in the case of a nonlinearMarkov process depend on 119901(119860 119905) as indicated in (37)The remaining conditional probabilities can be calculatedfrom 119879(119860 rarr 119861) = 1 minus 119879(119860 rarr 119860) and 119879(119861 rarr

119861) = 1 minus 119879(119861 rarr 119860) Model (37) has been examined inphysics social sciences and psychology as will be discussednext

32 Chaos in Probability Time-Discrete Case FollowingFrank [18] let us put 119879(119860 rarr 119860) = 119879(119861 rarr 119860) = 119879

0 Then

(37) becomes

119901 (119860 119905 + 1) = 1198790(119905 119901 (119860 119905)) (38)

Although (38) describes a stochastic process formally (38)exhibits the structure of a nonlinear deterministic mapsubjected to the constraint that the function 119879

0must be in

the interval [0 1] at any time step 119905 and for any occupationprobability 119901(119860) Iterative maps in general are known toexhibit chaos [19] Therefore (38) suggests that time-discretestrongly nonlinear Markov processes exhibit chaos as wellSuch strongly nonlinear chaotic processes exhibit chaos in theevolution of the occupation probabilities and the conditionalprobabilities For example in Frank [18] the logistic function1198790(119911) = 120572 sdot 119911(1 minus 119911) [19 20] was considered for which (38)

reads

119901 (119860 119905 + 1) = 120572119901 (119860 119905) [1 minus 119901 (119860 119905)] (39)

For 120572 in [0 4] the function 1198790(sdot) is in the interval [0 1] such

that (39) maps the occupation probability 119901(119860) from theinterval [0 1] to the interval [0 1] as required For parameters120572 ge 120572crit asymp 3570 [20] the map exhibits chaotic solutionsThat is 119901(119860 119905) evolves in a chaotic fashion However asstated previously not only the occupation probability exhibitschaos but the conditional probabilities 119879(119860 rarr 119860) =

119879(119861 rarr 119860) = 1198790exhibit a chaotic behavior as well which

was demonstrated explicitly by numerical simulations [18]Chaos in strongly nonlinear time-discrete Markov processesis not limited to the two-state models In fact the two-statemodel can be generalized to an 119873-state model In this case(38) becomes [18]

119901 (119896 119905 + 1) = 1198790(119905 119901 (119905)) (40)

for 119896 = 1 119873 minus 1 The occupation probability 119901(119873 119905)

is computed from the normalization condition Conse-quently 119879

0depends only on the occupation probabilities

119901(1 119905) 119901(119873 minus 1 119905) For example the two-dimensionalchaotic Bakermap [20] was described bymeans of (40) using119873 = 3 [18]

33 Social Sciences andGroupDecisionMaking Time-DiscreteCase Group decision making has frequently been describedby nonlinear stochastic processes [21ndash24]The reason for thisis that under certain circumstances individuals are likely tobe affected by the opinion that is presented by a group asa whole rather than by the opinion of individuals This so-called group opinion in turn is produced by the opinions anddecisions of the individual group members Consequentlythe decision making process of an individual is affected bythe sample average obtained from the opinions of the othersIf the groups can be considered as being relatively large thesample averagemay be replaced by the ensemble averageThedecision making behavior of an individual is then consideredas a realization of a stochastic process that is affected by theensemble average of the process In the context of votingbehavior the two-state model introduced in Section 31 was

8 ISRNMathematical Physics

applied [17]Themembers of theUSHouse of Representativeshad the choice to vote for or against passing the so-calledbailout bill (US House of Representatives USA 2008) Thebill failed to pass in the September 29 vote In a second voteon October 3 the bill finally passedThat is a critical numberof ldquoNordquo voters changed their opinions between September 29and October 3 The states 119896 = 1 and 119896 = 2 were labeled byldquo119884rdquo and ldquo119873rdquo representing ldquoYesrdquo and ldquoNordquo voters Accordingto model (37) the conditional probabilities 119879(119873 rarr 119884) and119879(119884 rarr 119884) describing a change in the opinion from ldquoNordquo toldquoYesrdquo and a consistent ldquoYesrdquo opinion were considered Usingthe notation of (37) the voting model proposed in Frank [18]reads

119901 (119884 119905 + 1)

= [119879 (119884 997888rarr 119884 119905 119901 (119884 119905)) minus 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))] 119901 (119884 119905)

+ 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))

(41)

The data available in the literature suggested that 119879(119884 rarr

119884) = 1 For the119879(119873 rarr 119884) the function119879(119873 rarr 119884) = 119860minus119861sdot

119901(119884 119905) was used to describe the impact of the group opinionon the voting behavior This relationship assumes that thepressure to change from a ldquoNordquo voter to a ldquoYesrdquo voter was highimmediately subsequent to the September 29 votes when theprobability 119901(119884 119905) at 119905 = Sept 29 was not high enough tomakethe bill pass In this context it should be mentioned that thefailure of the bill to pass on September 29 triggered a dramaticbreakdown of the US stock market That is the members ofthe House of Representatives who had voted with ldquoNordquo werenot only subjected to political pressure but also experiencedessential pressure from the economy For 119879(119884 rarr 119884) = 1

and 119879(119873 rarr 119884) = 119860 minus 119861119901(119884 119905) the stationary occupationprobability 119901(119884 st) is given by 119901(119884 st) = 119860119861 such that (41)eventually reads

119901 (119884 119905 + 1) = 119901 (119884 119905) + 119861 [119901 (119884 st) minus 119901 (119884 119905)] [1 minus 119901 (119884 119905)]

(42)

The stationary value 119901(119884 st) was estimated from the dataA detailed model-based analysis of the data showed thatthe members of the two main parties the Democratic andRepublican members were differently affected by the grouppressure

34 Psychology and Human Memory A pioneering experi-ment in humanmemory research is the free recall experimentin which participants are asked to recall items of a particularcategory [25 26] For example they are asked to recall animalnames Typically participants are instructed to recall as manyitems as possible and to do so as fast as possible and withoutrepetitionThe total number of recalled items is by definitiona monotonically increasing function of time Strikinglyfree recall involves clusters in which item subcategories arerecalled more or less as a whole For example when recallinganimal names a participant may recall a number of insectnames in succession From a theoretical point of view at anygiven time point during the free recall process the ability

to recall items in clusters depends on the size of the poolof items available for recalling In other words if there isa large pool of items that have not yet been recalled thenthere is a high chance that a whole item subcategory canbe recalled In line with this argument a stochastic modelfor free recall dynamics was developed in which the recallprocess depends on the size of the pool of items availablefor recall [27] More precisely time was parceled in discretetime intervals in which recall attempts are executed The aimwas to develop a model for the probability that a single itemhas been or has not yet been recalled at a given time step 119905To this end the two-state model of Section 31 was appliedAccordingly the states 119896 = 1 and 119896 = 2 correspond tobeing recalled or being not yet been recalled and are labeledby ldquo119877rdquo and ldquo119873rdquo respectively The two relevant conditionalprobabilities are119879(119877 rarr 119877) and119879(119873 rarr 119877) By the design ofthe experiment we have119879(119877 rarr 119877) = 1 Moreover as arguedpreviously if the pool of items that have not yet recalled islarge that is if the probability 119901(119873) that an item has notyet been recalled is high then the conditional probability119879(119873 rarr 119877) should be high aswell Likewise if the probability119901(119877) that an item has been recalled is high then 119879(119873 rarr 119877)

should be relatively low Therefore it was suggested to put119879(119873 rarr 119877) = 119861 sdot (1 minus 119901(119877 119905)) with 119861 gt 0 This functionimplies that the occupation probability 119901(119877 119905) converges tothe stationary fixed 119901(119877 st) = 1 at which 119879(119873 rarr 119877) = 0The full model can be obtained by substituting 119879(119877 rarr 119877)

and 119879(119873 rarr 119877) into (37) and reads

119901 (119877 119905 + 1) = 119901 (119877 119905) + 119861[1 minus 119901 (119877 119905)]2

(43)

Note that from a mathematical point of view this modelis a special case of model (42) for voting behavior (hintput 119901(119884 st) = 1 in (42)) Since the model describes thesingle item recall probability the model must be scaled tothe number of items recalled by any given participant Tothis end a parameter 119862 describing the maximal number ofitems for recall was introduced The unknown parameters119861 and 119862 were estimated for free recall data collected in astudy by Rhodes and Turvey [28] and the sample meanvalues 119861 = 0009 (SD = 0005) and 119862 = 307 (SD = 138)were found for a sample of eight participants [27] Note that(43) predicts that the fixed point 119901(119877 st) = 1 will neverbe reached in a finite number of time steps Consequentlythe model predicts that the total number of recalled items119872 at the end of the experiment is smaller than the numberof available items for recall This prediction was confirmedby the data Moreover the model-based analysis revealed astatistically significant positive correlation between 119872 and119862 for the participant sample studied in Frank and Rhodes[27]

35 Biology and Evolutionary Population Dynamics A pri-mary concern of evolutionary population dynamics is theunderstanding of the origin of life Among the many modelsthat have been proposed in this context the studies byCrutchfield and Gornerup [29] and Gornerup and Crutch-field [30] are of interest for our review In these studiesa number of 119873 macromolecules with different functional

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 5

form of the master equation If the integral is evaluated for asmall time interval Δ119905 = 119905 minus 119905

1015840

rarr 0 then the integral versionexhibits the structure of a Markov chain (7) with

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905)) = Δ119905 sdot 119908 (119894 997888rarr 119896 119905 119901 (119905)) 119894 = 119896

119879 (119896 997888rarr 119896 119905 Δ119905 119901 (119905)) = 1 minus Δ119905

119873

sum

119895=1

119908 (119896 997888rarr 119895 119905 119901 (119905))

(18)

which can be written in a more concise form like

119879 (119894 997888rarr 119896 119905 Δ119905 119901 (119905))

= Δ119905 119908 (119894 997888rarr 119896 119905 119901 (119905)) + 120575 (119894 119896)

sdot [

[

1 minus Δ119905

119873

sum

119895=1

119908 (119894 997888rarr 119895 119905 119901 (119905))]

]

(19)

Note again that we will use the convention that the diagonalelements 119908(119896 rarr 119896) of the transition rates equal zero whichimplies that for the diagonal elements 119879(119896 rarr 119896) only thesecond term in (19) matters as stated in (18) For the sake ofcompleteness note that the differential version of the masterequation (17) reads

119889

119889119905119901 (119896 119905) =

119873

sum

119894=1

119908 (119894 997888rarr 119896 119905 119901 (119905)) 119901 (119894 119905)

minus 119901 (119896 119905)

119873

sum

119894=1

119908 (119896 997888rarr 119894 119905 119901 (119905))

(20)

Just as in Section 22 in order to describe the four classes ofstochastic processes listed in Table 1 on an equal footing weconsider an alternative description of the process defined by(20) Accordingly we introduce the flow functionΦ(119905 120576) thatmaps initial states119883(0) defined on the integer set 1 119873 tostates119883(119905)

119883 (119905) = Φ (119883 (0) 120576) (21)

The variable 120576 is a random variable that is uniformly dis-tributed on the interval [0 1] at every time point 119905 and isstatistically independently distributed across time In view ofthe fact that the time-discrete version of the master equationexhibits the structure of a Markov chain trajectories definedby the flow function can be regarded as time-continuouscounterparts to the trajectories of Markov chains as definedby (1) and (8)ndash(11)This analogy can be exploited to constructiteratively the flow function Φ in the limiting case Δ119905 rarr 0

like

Φ 119883 (119905 + Δ119905) = 119891 (119883 (119905) 119905 Δ119905 119901 (119905) 120576 (119905)) (22)

with

119891 (119883 (119905) = 119894 119905 Δ119905 119901 (119905) 120576 (119905))

=

119873

sum

119896=1

119896 sdot 119868 (119894 997888rarr 119896 119905 Δ119905 119901 (119905) 120576 (119905))

(23)

The iterative map 119891(sdot) involves the indicator function

119868 (119894 997888rarr 119896 120576 119905 Δ119905 119901) = 1 for 120576 isin [119878 (119894 119896 minus 1) 119878 (119894 119896))

0 otherwise(24)

and the cumulative transition probabilities

119878 (119894 119896) =

119896

sum

119895=1

119879 (119894 997888rarr 119895 119905 Δ119905 119901 (119905)) (25)

Equations (1) (18) and (22)ndash(25) provide a closed approxi-mate description of the trajectories of interest By definitionthe description becomes exact in the limiting case of infinitelysmall time intervals Δ119905 Note that the conditional (transition)probabilities 119879(sdot) defined by (18) are in the interval [0 1]provided that Δ119905 is chosen sufficiently small In the limitingcase Δ119905 rarr 0 this is always the case assuming that the rates119908(sdot) are bounded from above Note also that the probabilities119879(sdot) defined by (18) are normalized to 1 Furthermore from(18) and (22)ndash(25) it follows that 119891(sdot) = 119894 holds for Δ119905 = 0That is the iterativemap (22) describes transitions from states119883(119905) = 119894 to states119883(119905+Δ119905) in the small time intervalΔ119905 Mostimportantly according to (22) the evolution of realizationsdepends on the occupation probability vector 119901(119905) at everytime point 119905 Therefore processes defined by (1) (18) and(22)ndash(25) belong to the class of strongly nonlinear stochasticprocesses (see the definition in Section 12)

25 Strongly Nonlinear Markov Diffusion Processes Defined byStrongly Nonlinear Langevin Equations and Strongly Nonline-ar Fokker-Planck Equations Realizations of time-continuousstochastic processes evolving on119873-dimensional continuousstate spaces are described by time-dependent state vectors119883(119905) with time-dependent components 119883

1(119905) 119883

119873(119905) At

every time point 119905 the vector 119883(119905) is distributed accordingto the probability density 119875(119909 119905) defined by (12) where thevariable 119909 is the state vector with coordinates 119909

1 119909

119873

Strongly nonlinear Markov diffusion processes have realiza-tions (or trajectories) defined in the limiting case of infinitelysmall time intervals Δ119905 rarr 0 by strongly nonlinear stochasticdifference equations [9] of the form

119883 (119905 + Δ119905) = 119883 (119905) + Δ119905ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)])

+ radic2Δ119905119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot 120576 (119905)

(26)

Carrying out the limiting procedure (26) can be writtenin a more concise way in terms of a so-called Ito-Langevinequation

119889

119889119905119883 (119905) = ℎ (119883 (119905) 119905 120579

119883(119905)[119875 (119909 119905)])

+ 119892 (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) sdot Γ (119905)

(27)

Just as in the case of the stochastic iterative maps discussed inSection 23 (26) and (27) exhibit two different components

6 ISRNMathematical Physics

First the vector-valued function ℎ(sdot) occurring in (26) and(27) is the drift function introduced in Section 23 thataccounts for the deterministic evolution of a process Secondthe expressions 119892 sdot 120576 and 119892 sdot Γ respectively describe avector-valued fluctuating force The variable 120576(119905) is the 119873-dimensional random variable introduced in Section 23 Thefunction Γ(119905) in the Ito-Langevin equation (27) is the coun-terpart to the random vector 120576(119905) in (26) and corresponds to aso-called vector-valued Langevin force [4 6] More preciselyeach component Γ

119896(119905) is a Langevin force We will consider

Langevin forces normalized with respect to a magnitude of 2like

lim119877rarrinfin

1

119877Γ119903

119895(119905) Γ119903

119896(119904) = 2120575 (119895 119896) 120575 (119905 minus 119904) (28)

where Γ119903

119896(119905) are realizations of the 119896th component The

variable 119892 is the 119873 times 119873 weight matrix introduced inSection 23

The stochastic process defined by the Ito-Langevin equa-tion (27) can equivalently be expressed bymeans of a stronglynonlinear Fokker-Planck equation

120597

120597119905119875 (119909 119905) = minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119875 (119909 119905)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119875 (119909 119905)]

(29)

The evolution equation for 119875(119909 119905) is nonlinear for 119875(119909 119905)In contrast the corresponding evolution equation for theconditional probability density 119879(119909 119905 | 119910 119904) with 119905 gt 119904 islinear with respect to 119879(119909 119905 | 119910 119904) and reads

120597

120597119905119879 (119909 119905 | 119910 119904)

= minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119879 (119909 119905 | 119910 119904)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119879 (119909 119905 | 119910 119904)]

(30)

For details see Frank [9] A complete description of thestochastic process can be obtained either from the realiza-tions defined by (26) and (27) or from the evolution equations(29) and (30) for 119875(119909 119905) and 119879(119909 119905 | 119910 119904) The expression

119863119894119895(119909 119905 120579

119909[119875 (sdot 119905)]) =

119873

sum

119896=1

119892119894119896(sdot) 119892119895119896(sdot) (31)

occurring in (29) and (30) is the diffusion coefficient of theprocess

A fundamental example of a strongly nonlinear Markovdiffusion process is a generalized Ornstein-Uhlenbeck pro-cess involving a mean field force ℎMF proportional to the

mean ⟨119883(119905)⟩ of the process [9 15 16] For 119873 = 1 thestochastic difference equation reads

119883 (119905 + Δ119905) = 119883 (119905) minus Δ119905 [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + radic2Δ119905119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(32)

with 119892 ge 0 Typically the case119860 119861 gt 0 is considered For (32)the strongly nonlinear Langevin equation reads

119889

119889119905119883 (119905) = minus [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + 119892Γ (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(33)

Finally the corresponding strongly nonlinear Fokker-Planckequation reads

120597

120597119905119875 (119909 119905) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119875 (119909 119905)]

+ 1198631205972

1205971199092119875 (119909 119905)

120597

120597119905119879 (119909 119905 | 119910 119904) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119879 (119909 119905 | 119910 119904)]

+ 1198631205972

1205971199092119879 (119909 119905 | 119910 119904)

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

119863 = 1198922

(34)

The generalized Ornstein-Uhlenbeck process in its form asstochastic difference equation (32) corresponds to a specialcase of the generalized autoregressive model of order 1defined by (16) Model (34) has been studied in the contextof the Shimizu-Yamada model We will return to this issuelater

26 On a General Stochastic Evolution Equation for StronglyNonlinear Markov Processes In the previous sections weconsidered some benchmark examples of strongly nonlinearstochastic processes belonging to the four classes of processesdefined in Table 1 All processes considered in Sections 22to 25 satisfy the Markov property A more general formaldescription of strongly nonlinear Markov processes thatholds for all four classes reads

119883 (119905 + 120585) = 119891 (119883 (119905) 119905 120585 120579119883(119905)

[119875 (119909 119905)] 120576 (119905)) (35)

The additional variable 120585 defines the characteristic jumpinginterval of the process under consideration and can be used todistinguish between time-discrete and time-continuous pro-cesses For 120585 = 1 we are dealing with time-discrete stochastic

ISRNMathematical Physics 7

processes In the case of processes defined on discrete statespaces 119875(119909 119905) is related to the occupation probability 119901(119896 119905)by means of (5) and (6) and (35) reduces to the special caseof (8) of the Markov chain model For processes defined oncontinuous state spaces and 120585 = 1 the general form (35)becomes (13) in the case of stochastic processes driven bynonparametric white noise In the limiting case 120585 rarr 0 weare dealing with time-continuous stochastic processes In thiscase the function 119891(sdot) possesses the property 119891(sdot) = 119883(119905)

for 120585 = 0 For processes that evolve on discrete state spacesand satisfy a nonlinear master equation it follows that (35)becomes the iterative map (22) provided that the argument119875(119909 119905) in 119891(sdot) is expressed in terms of the occupationprobability 119901(119896 119905) see (6) In contrast for processes evolvingon continuous state spaces and satisfying strongly nonlinearIto-Langevin equations the general form (35) reduces to thestochastic difference equation (26) when we consider thelimiting case 120585 rarr 0

3 Time-Discrete Strongly Nonlinear StochasticProcesses on Discrete State Spaces

31 The Strongly Nonlinear Two-State Markov Chain ModelA fundamental strongly nonlinear stochastic process is thetwo-state nonlinear Markov chain process [17] For the sakeof convenience the states 119896 = 1 and 119896 = 2will be labeled withldquo119860rdquo and ldquo119861rdquo Transitions from119860 to119860119860 to 119861 119861 to119860 and 119861 to119861 occur with conditional probabilities 119879(119860 rarr 119860) 119879(119860 rarr

119861) 119879(119861 rarr 119860) and 119879(119861 rarr 119861) respectively Accordinglythe nonlinear Markov chain equation (7) reads

119901 (119860 119905 + 1) = 119879 (119860 997888rarr 119860 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119905)) 119901 (119861 119905)

119901 (119861 119905 + 1) = 119879 (119860 997888rarr 119861 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119861 119905 119901 (119905)) 119901 (119861 119905)

(36)

Taking the normalization condition 119901(119860) + 119901(119861) = 1 intoconsideration (36) can equivalently be expressed by

119901 (119860 119905 + 1)

= [119879 (119860 997888rarr 119860 119905 119901 (119860 119905)) minus 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))] 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))

(37)

which is a closed nonlinear equation in 119901(119860 119905) Model (37)involves two functions only the conditional probabilities119879(119860 rarr 119860) and 119879(119861 rarr 119860) which can be chosenindependently from each other and in the case of a nonlinearMarkov process depend on 119901(119860 119905) as indicated in (37)The remaining conditional probabilities can be calculatedfrom 119879(119860 rarr 119861) = 1 minus 119879(119860 rarr 119860) and 119879(119861 rarr

119861) = 1 minus 119879(119861 rarr 119860) Model (37) has been examined inphysics social sciences and psychology as will be discussednext

32 Chaos in Probability Time-Discrete Case FollowingFrank [18] let us put 119879(119860 rarr 119860) = 119879(119861 rarr 119860) = 119879

0 Then

(37) becomes

119901 (119860 119905 + 1) = 1198790(119905 119901 (119860 119905)) (38)

Although (38) describes a stochastic process formally (38)exhibits the structure of a nonlinear deterministic mapsubjected to the constraint that the function 119879

0must be in

the interval [0 1] at any time step 119905 and for any occupationprobability 119901(119860) Iterative maps in general are known toexhibit chaos [19] Therefore (38) suggests that time-discretestrongly nonlinear Markov processes exhibit chaos as wellSuch strongly nonlinear chaotic processes exhibit chaos in theevolution of the occupation probabilities and the conditionalprobabilities For example in Frank [18] the logistic function1198790(119911) = 120572 sdot 119911(1 minus 119911) [19 20] was considered for which (38)

reads

119901 (119860 119905 + 1) = 120572119901 (119860 119905) [1 minus 119901 (119860 119905)] (39)

For 120572 in [0 4] the function 1198790(sdot) is in the interval [0 1] such

that (39) maps the occupation probability 119901(119860) from theinterval [0 1] to the interval [0 1] as required For parameters120572 ge 120572crit asymp 3570 [20] the map exhibits chaotic solutionsThat is 119901(119860 119905) evolves in a chaotic fashion However asstated previously not only the occupation probability exhibitschaos but the conditional probabilities 119879(119860 rarr 119860) =

119879(119861 rarr 119860) = 1198790exhibit a chaotic behavior as well which

was demonstrated explicitly by numerical simulations [18]Chaos in strongly nonlinear time-discrete Markov processesis not limited to the two-state models In fact the two-statemodel can be generalized to an 119873-state model In this case(38) becomes [18]

119901 (119896 119905 + 1) = 1198790(119905 119901 (119905)) (40)

for 119896 = 1 119873 minus 1 The occupation probability 119901(119873 119905)

is computed from the normalization condition Conse-quently 119879

0depends only on the occupation probabilities

119901(1 119905) 119901(119873 minus 1 119905) For example the two-dimensionalchaotic Bakermap [20] was described bymeans of (40) using119873 = 3 [18]

33 Social Sciences andGroupDecisionMaking Time-DiscreteCase Group decision making has frequently been describedby nonlinear stochastic processes [21ndash24]The reason for thisis that under certain circumstances individuals are likely tobe affected by the opinion that is presented by a group asa whole rather than by the opinion of individuals This so-called group opinion in turn is produced by the opinions anddecisions of the individual group members Consequentlythe decision making process of an individual is affected bythe sample average obtained from the opinions of the othersIf the groups can be considered as being relatively large thesample averagemay be replaced by the ensemble averageThedecision making behavior of an individual is then consideredas a realization of a stochastic process that is affected by theensemble average of the process In the context of votingbehavior the two-state model introduced in Section 31 was

8 ISRNMathematical Physics

applied [17]Themembers of theUSHouse of Representativeshad the choice to vote for or against passing the so-calledbailout bill (US House of Representatives USA 2008) Thebill failed to pass in the September 29 vote In a second voteon October 3 the bill finally passedThat is a critical numberof ldquoNordquo voters changed their opinions between September 29and October 3 The states 119896 = 1 and 119896 = 2 were labeled byldquo119884rdquo and ldquo119873rdquo representing ldquoYesrdquo and ldquoNordquo voters Accordingto model (37) the conditional probabilities 119879(119873 rarr 119884) and119879(119884 rarr 119884) describing a change in the opinion from ldquoNordquo toldquoYesrdquo and a consistent ldquoYesrdquo opinion were considered Usingthe notation of (37) the voting model proposed in Frank [18]reads

119901 (119884 119905 + 1)

= [119879 (119884 997888rarr 119884 119905 119901 (119884 119905)) minus 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))] 119901 (119884 119905)

+ 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))

(41)

The data available in the literature suggested that 119879(119884 rarr

119884) = 1 For the119879(119873 rarr 119884) the function119879(119873 rarr 119884) = 119860minus119861sdot

119901(119884 119905) was used to describe the impact of the group opinionon the voting behavior This relationship assumes that thepressure to change from a ldquoNordquo voter to a ldquoYesrdquo voter was highimmediately subsequent to the September 29 votes when theprobability 119901(119884 119905) at 119905 = Sept 29 was not high enough tomakethe bill pass In this context it should be mentioned that thefailure of the bill to pass on September 29 triggered a dramaticbreakdown of the US stock market That is the members ofthe House of Representatives who had voted with ldquoNordquo werenot only subjected to political pressure but also experiencedessential pressure from the economy For 119879(119884 rarr 119884) = 1

and 119879(119873 rarr 119884) = 119860 minus 119861119901(119884 119905) the stationary occupationprobability 119901(119884 st) is given by 119901(119884 st) = 119860119861 such that (41)eventually reads

119901 (119884 119905 + 1) = 119901 (119884 119905) + 119861 [119901 (119884 st) minus 119901 (119884 119905)] [1 minus 119901 (119884 119905)]

(42)

The stationary value 119901(119884 st) was estimated from the dataA detailed model-based analysis of the data showed thatthe members of the two main parties the Democratic andRepublican members were differently affected by the grouppressure

34 Psychology and Human Memory A pioneering experi-ment in humanmemory research is the free recall experimentin which participants are asked to recall items of a particularcategory [25 26] For example they are asked to recall animalnames Typically participants are instructed to recall as manyitems as possible and to do so as fast as possible and withoutrepetitionThe total number of recalled items is by definitiona monotonically increasing function of time Strikinglyfree recall involves clusters in which item subcategories arerecalled more or less as a whole For example when recallinganimal names a participant may recall a number of insectnames in succession From a theoretical point of view at anygiven time point during the free recall process the ability

to recall items in clusters depends on the size of the poolof items available for recalling In other words if there isa large pool of items that have not yet been recalled thenthere is a high chance that a whole item subcategory canbe recalled In line with this argument a stochastic modelfor free recall dynamics was developed in which the recallprocess depends on the size of the pool of items availablefor recall [27] More precisely time was parceled in discretetime intervals in which recall attempts are executed The aimwas to develop a model for the probability that a single itemhas been or has not yet been recalled at a given time step 119905To this end the two-state model of Section 31 was appliedAccordingly the states 119896 = 1 and 119896 = 2 correspond tobeing recalled or being not yet been recalled and are labeledby ldquo119877rdquo and ldquo119873rdquo respectively The two relevant conditionalprobabilities are119879(119877 rarr 119877) and119879(119873 rarr 119877) By the design ofthe experiment we have119879(119877 rarr 119877) = 1 Moreover as arguedpreviously if the pool of items that have not yet recalled islarge that is if the probability 119901(119873) that an item has notyet been recalled is high then the conditional probability119879(119873 rarr 119877) should be high aswell Likewise if the probability119901(119877) that an item has been recalled is high then 119879(119873 rarr 119877)

should be relatively low Therefore it was suggested to put119879(119873 rarr 119877) = 119861 sdot (1 minus 119901(119877 119905)) with 119861 gt 0 This functionimplies that the occupation probability 119901(119877 119905) converges tothe stationary fixed 119901(119877 st) = 1 at which 119879(119873 rarr 119877) = 0The full model can be obtained by substituting 119879(119877 rarr 119877)

and 119879(119873 rarr 119877) into (37) and reads

119901 (119877 119905 + 1) = 119901 (119877 119905) + 119861[1 minus 119901 (119877 119905)]2

(43)

Note that from a mathematical point of view this modelis a special case of model (42) for voting behavior (hintput 119901(119884 st) = 1 in (42)) Since the model describes thesingle item recall probability the model must be scaled tothe number of items recalled by any given participant Tothis end a parameter 119862 describing the maximal number ofitems for recall was introduced The unknown parameters119861 and 119862 were estimated for free recall data collected in astudy by Rhodes and Turvey [28] and the sample meanvalues 119861 = 0009 (SD = 0005) and 119862 = 307 (SD = 138)were found for a sample of eight participants [27] Note that(43) predicts that the fixed point 119901(119877 st) = 1 will neverbe reached in a finite number of time steps Consequentlythe model predicts that the total number of recalled items119872 at the end of the experiment is smaller than the numberof available items for recall This prediction was confirmedby the data Moreover the model-based analysis revealed astatistically significant positive correlation between 119872 and119862 for the participant sample studied in Frank and Rhodes[27]

35 Biology and Evolutionary Population Dynamics A pri-mary concern of evolutionary population dynamics is theunderstanding of the origin of life Among the many modelsthat have been proposed in this context the studies byCrutchfield and Gornerup [29] and Gornerup and Crutch-field [30] are of interest for our review In these studiesa number of 119873 macromolecules with different functional

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

6 ISRNMathematical Physics

First the vector-valued function ℎ(sdot) occurring in (26) and(27) is the drift function introduced in Section 23 thataccounts for the deterministic evolution of a process Secondthe expressions 119892 sdot 120576 and 119892 sdot Γ respectively describe avector-valued fluctuating force The variable 120576(119905) is the 119873-dimensional random variable introduced in Section 23 Thefunction Γ(119905) in the Ito-Langevin equation (27) is the coun-terpart to the random vector 120576(119905) in (26) and corresponds to aso-called vector-valued Langevin force [4 6] More preciselyeach component Γ

119896(119905) is a Langevin force We will consider

Langevin forces normalized with respect to a magnitude of 2like

lim119877rarrinfin

1

119877Γ119903

119895(119905) Γ119903

119896(119904) = 2120575 (119895 119896) 120575 (119905 minus 119904) (28)

where Γ119903

119896(119905) are realizations of the 119896th component The

variable 119892 is the 119873 times 119873 weight matrix introduced inSection 23

The stochastic process defined by the Ito-Langevin equa-tion (27) can equivalently be expressed bymeans of a stronglynonlinear Fokker-Planck equation

120597

120597119905119875 (119909 119905) = minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119875 (119909 119905)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119875 (119909 119905)]

(29)

The evolution equation for 119875(119909 119905) is nonlinear for 119875(119909 119905)In contrast the corresponding evolution equation for theconditional probability density 119879(119909 119905 | 119910 119904) with 119905 gt 119904 islinear with respect to 119879(119909 119905 | 119910 119904) and reads

120597

120597119905119879 (119909 119905 | 119910 119904)

= minus

119873

sum

119896=1

120597

120597119909119896

[ℎ119896(119909 119905 120579

119909[119875 (sdot 119905)]) 119879 (119909 119905 | 119910 119904)]

+

119873

sum

119894=1119895=1119896=1

1205972

120597119909119894120597119909119895

[119892119894119896(sdot) 119892119895119896(sdot) 119879 (119909 119905 | 119910 119904)]

(30)

For details see Frank [9] A complete description of thestochastic process can be obtained either from the realiza-tions defined by (26) and (27) or from the evolution equations(29) and (30) for 119875(119909 119905) and 119879(119909 119905 | 119910 119904) The expression

119863119894119895(119909 119905 120579

119909[119875 (sdot 119905)]) =

119873

sum

119896=1

119892119894119896(sdot) 119892119895119896(sdot) (31)

occurring in (29) and (30) is the diffusion coefficient of theprocess

A fundamental example of a strongly nonlinear Markovdiffusion process is a generalized Ornstein-Uhlenbeck pro-cess involving a mean field force ℎMF proportional to the

mean ⟨119883(119905)⟩ of the process [9 15 16] For 119873 = 1 thestochastic difference equation reads

119883 (119905 + Δ119905) = 119883 (119905) minus Δ119905 [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + radic2Δ119905119892120576 (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(32)

with 119892 ge 0 Typically the case119860 119861 gt 0 is considered For (32)the strongly nonlinear Langevin equation reads

119889

119889119905119883 (119905) = minus [119860119883 (119905) + 119861 ⟨119883 (119905)⟩] + 119892Γ (119905)

⟨119883 (119905)⟩ = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(33)

Finally the corresponding strongly nonlinear Fokker-Planckequation reads

120597

120597119905119875 (119909 119905) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119875 (119909 119905)]

+ 1198631205972

1205971199092119875 (119909 119905)

120597

120597119905119879 (119909 119905 | 119910 119904) =

120597

120597119909[(119860119909 + 119861119872(119905)) 119879 (119909 119905 | 119910 119904)]

+ 1198631205972

1205971199092119879 (119909 119905 | 119910 119904)

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

119863 = 1198922

(34)

The generalized Ornstein-Uhlenbeck process in its form asstochastic difference equation (32) corresponds to a specialcase of the generalized autoregressive model of order 1defined by (16) Model (34) has been studied in the contextof the Shimizu-Yamada model We will return to this issuelater

26 On a General Stochastic Evolution Equation for StronglyNonlinear Markov Processes In the previous sections weconsidered some benchmark examples of strongly nonlinearstochastic processes belonging to the four classes of processesdefined in Table 1 All processes considered in Sections 22to 25 satisfy the Markov property A more general formaldescription of strongly nonlinear Markov processes thatholds for all four classes reads

119883 (119905 + 120585) = 119891 (119883 (119905) 119905 120585 120579119883(119905)

[119875 (119909 119905)] 120576 (119905)) (35)

The additional variable 120585 defines the characteristic jumpinginterval of the process under consideration and can be used todistinguish between time-discrete and time-continuous pro-cesses For 120585 = 1 we are dealing with time-discrete stochastic

ISRNMathematical Physics 7

processes In the case of processes defined on discrete statespaces 119875(119909 119905) is related to the occupation probability 119901(119896 119905)by means of (5) and (6) and (35) reduces to the special caseof (8) of the Markov chain model For processes defined oncontinuous state spaces and 120585 = 1 the general form (35)becomes (13) in the case of stochastic processes driven bynonparametric white noise In the limiting case 120585 rarr 0 weare dealing with time-continuous stochastic processes In thiscase the function 119891(sdot) possesses the property 119891(sdot) = 119883(119905)

for 120585 = 0 For processes that evolve on discrete state spacesand satisfy a nonlinear master equation it follows that (35)becomes the iterative map (22) provided that the argument119875(119909 119905) in 119891(sdot) is expressed in terms of the occupationprobability 119901(119896 119905) see (6) In contrast for processes evolvingon continuous state spaces and satisfying strongly nonlinearIto-Langevin equations the general form (35) reduces to thestochastic difference equation (26) when we consider thelimiting case 120585 rarr 0

3 Time-Discrete Strongly Nonlinear StochasticProcesses on Discrete State Spaces

31 The Strongly Nonlinear Two-State Markov Chain ModelA fundamental strongly nonlinear stochastic process is thetwo-state nonlinear Markov chain process [17] For the sakeof convenience the states 119896 = 1 and 119896 = 2will be labeled withldquo119860rdquo and ldquo119861rdquo Transitions from119860 to119860119860 to 119861 119861 to119860 and 119861 to119861 occur with conditional probabilities 119879(119860 rarr 119860) 119879(119860 rarr

119861) 119879(119861 rarr 119860) and 119879(119861 rarr 119861) respectively Accordinglythe nonlinear Markov chain equation (7) reads

119901 (119860 119905 + 1) = 119879 (119860 997888rarr 119860 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119905)) 119901 (119861 119905)

119901 (119861 119905 + 1) = 119879 (119860 997888rarr 119861 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119861 119905 119901 (119905)) 119901 (119861 119905)

(36)

Taking the normalization condition 119901(119860) + 119901(119861) = 1 intoconsideration (36) can equivalently be expressed by

119901 (119860 119905 + 1)

= [119879 (119860 997888rarr 119860 119905 119901 (119860 119905)) minus 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))] 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))

(37)

which is a closed nonlinear equation in 119901(119860 119905) Model (37)involves two functions only the conditional probabilities119879(119860 rarr 119860) and 119879(119861 rarr 119860) which can be chosenindependently from each other and in the case of a nonlinearMarkov process depend on 119901(119860 119905) as indicated in (37)The remaining conditional probabilities can be calculatedfrom 119879(119860 rarr 119861) = 1 minus 119879(119860 rarr 119860) and 119879(119861 rarr

119861) = 1 minus 119879(119861 rarr 119860) Model (37) has been examined inphysics social sciences and psychology as will be discussednext

32 Chaos in Probability Time-Discrete Case FollowingFrank [18] let us put 119879(119860 rarr 119860) = 119879(119861 rarr 119860) = 119879

0 Then

(37) becomes

119901 (119860 119905 + 1) = 1198790(119905 119901 (119860 119905)) (38)

Although (38) describes a stochastic process formally (38)exhibits the structure of a nonlinear deterministic mapsubjected to the constraint that the function 119879

0must be in

the interval [0 1] at any time step 119905 and for any occupationprobability 119901(119860) Iterative maps in general are known toexhibit chaos [19] Therefore (38) suggests that time-discretestrongly nonlinear Markov processes exhibit chaos as wellSuch strongly nonlinear chaotic processes exhibit chaos in theevolution of the occupation probabilities and the conditionalprobabilities For example in Frank [18] the logistic function1198790(119911) = 120572 sdot 119911(1 minus 119911) [19 20] was considered for which (38)

reads

119901 (119860 119905 + 1) = 120572119901 (119860 119905) [1 minus 119901 (119860 119905)] (39)

For 120572 in [0 4] the function 1198790(sdot) is in the interval [0 1] such

that (39) maps the occupation probability 119901(119860) from theinterval [0 1] to the interval [0 1] as required For parameters120572 ge 120572crit asymp 3570 [20] the map exhibits chaotic solutionsThat is 119901(119860 119905) evolves in a chaotic fashion However asstated previously not only the occupation probability exhibitschaos but the conditional probabilities 119879(119860 rarr 119860) =

119879(119861 rarr 119860) = 1198790exhibit a chaotic behavior as well which

was demonstrated explicitly by numerical simulations [18]Chaos in strongly nonlinear time-discrete Markov processesis not limited to the two-state models In fact the two-statemodel can be generalized to an 119873-state model In this case(38) becomes [18]

119901 (119896 119905 + 1) = 1198790(119905 119901 (119905)) (40)

for 119896 = 1 119873 minus 1 The occupation probability 119901(119873 119905)

is computed from the normalization condition Conse-quently 119879

0depends only on the occupation probabilities

119901(1 119905) 119901(119873 minus 1 119905) For example the two-dimensionalchaotic Bakermap [20] was described bymeans of (40) using119873 = 3 [18]

33 Social Sciences andGroupDecisionMaking Time-DiscreteCase Group decision making has frequently been describedby nonlinear stochastic processes [21ndash24]The reason for thisis that under certain circumstances individuals are likely tobe affected by the opinion that is presented by a group asa whole rather than by the opinion of individuals This so-called group opinion in turn is produced by the opinions anddecisions of the individual group members Consequentlythe decision making process of an individual is affected bythe sample average obtained from the opinions of the othersIf the groups can be considered as being relatively large thesample averagemay be replaced by the ensemble averageThedecision making behavior of an individual is then consideredas a realization of a stochastic process that is affected by theensemble average of the process In the context of votingbehavior the two-state model introduced in Section 31 was

8 ISRNMathematical Physics

applied [17]Themembers of theUSHouse of Representativeshad the choice to vote for or against passing the so-calledbailout bill (US House of Representatives USA 2008) Thebill failed to pass in the September 29 vote In a second voteon October 3 the bill finally passedThat is a critical numberof ldquoNordquo voters changed their opinions between September 29and October 3 The states 119896 = 1 and 119896 = 2 were labeled byldquo119884rdquo and ldquo119873rdquo representing ldquoYesrdquo and ldquoNordquo voters Accordingto model (37) the conditional probabilities 119879(119873 rarr 119884) and119879(119884 rarr 119884) describing a change in the opinion from ldquoNordquo toldquoYesrdquo and a consistent ldquoYesrdquo opinion were considered Usingthe notation of (37) the voting model proposed in Frank [18]reads

119901 (119884 119905 + 1)

= [119879 (119884 997888rarr 119884 119905 119901 (119884 119905)) minus 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))] 119901 (119884 119905)

+ 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))

(41)

The data available in the literature suggested that 119879(119884 rarr

119884) = 1 For the119879(119873 rarr 119884) the function119879(119873 rarr 119884) = 119860minus119861sdot

119901(119884 119905) was used to describe the impact of the group opinionon the voting behavior This relationship assumes that thepressure to change from a ldquoNordquo voter to a ldquoYesrdquo voter was highimmediately subsequent to the September 29 votes when theprobability 119901(119884 119905) at 119905 = Sept 29 was not high enough tomakethe bill pass In this context it should be mentioned that thefailure of the bill to pass on September 29 triggered a dramaticbreakdown of the US stock market That is the members ofthe House of Representatives who had voted with ldquoNordquo werenot only subjected to political pressure but also experiencedessential pressure from the economy For 119879(119884 rarr 119884) = 1

and 119879(119873 rarr 119884) = 119860 minus 119861119901(119884 119905) the stationary occupationprobability 119901(119884 st) is given by 119901(119884 st) = 119860119861 such that (41)eventually reads

119901 (119884 119905 + 1) = 119901 (119884 119905) + 119861 [119901 (119884 st) minus 119901 (119884 119905)] [1 minus 119901 (119884 119905)]

(42)

The stationary value 119901(119884 st) was estimated from the dataA detailed model-based analysis of the data showed thatthe members of the two main parties the Democratic andRepublican members were differently affected by the grouppressure

34 Psychology and Human Memory A pioneering experi-ment in humanmemory research is the free recall experimentin which participants are asked to recall items of a particularcategory [25 26] For example they are asked to recall animalnames Typically participants are instructed to recall as manyitems as possible and to do so as fast as possible and withoutrepetitionThe total number of recalled items is by definitiona monotonically increasing function of time Strikinglyfree recall involves clusters in which item subcategories arerecalled more or less as a whole For example when recallinganimal names a participant may recall a number of insectnames in succession From a theoretical point of view at anygiven time point during the free recall process the ability

to recall items in clusters depends on the size of the poolof items available for recalling In other words if there isa large pool of items that have not yet been recalled thenthere is a high chance that a whole item subcategory canbe recalled In line with this argument a stochastic modelfor free recall dynamics was developed in which the recallprocess depends on the size of the pool of items availablefor recall [27] More precisely time was parceled in discretetime intervals in which recall attempts are executed The aimwas to develop a model for the probability that a single itemhas been or has not yet been recalled at a given time step 119905To this end the two-state model of Section 31 was appliedAccordingly the states 119896 = 1 and 119896 = 2 correspond tobeing recalled or being not yet been recalled and are labeledby ldquo119877rdquo and ldquo119873rdquo respectively The two relevant conditionalprobabilities are119879(119877 rarr 119877) and119879(119873 rarr 119877) By the design ofthe experiment we have119879(119877 rarr 119877) = 1 Moreover as arguedpreviously if the pool of items that have not yet recalled islarge that is if the probability 119901(119873) that an item has notyet been recalled is high then the conditional probability119879(119873 rarr 119877) should be high aswell Likewise if the probability119901(119877) that an item has been recalled is high then 119879(119873 rarr 119877)

should be relatively low Therefore it was suggested to put119879(119873 rarr 119877) = 119861 sdot (1 minus 119901(119877 119905)) with 119861 gt 0 This functionimplies that the occupation probability 119901(119877 119905) converges tothe stationary fixed 119901(119877 st) = 1 at which 119879(119873 rarr 119877) = 0The full model can be obtained by substituting 119879(119877 rarr 119877)

and 119879(119873 rarr 119877) into (37) and reads

119901 (119877 119905 + 1) = 119901 (119877 119905) + 119861[1 minus 119901 (119877 119905)]2

(43)

Note that from a mathematical point of view this modelis a special case of model (42) for voting behavior (hintput 119901(119884 st) = 1 in (42)) Since the model describes thesingle item recall probability the model must be scaled tothe number of items recalled by any given participant Tothis end a parameter 119862 describing the maximal number ofitems for recall was introduced The unknown parameters119861 and 119862 were estimated for free recall data collected in astudy by Rhodes and Turvey [28] and the sample meanvalues 119861 = 0009 (SD = 0005) and 119862 = 307 (SD = 138)were found for a sample of eight participants [27] Note that(43) predicts that the fixed point 119901(119877 st) = 1 will neverbe reached in a finite number of time steps Consequentlythe model predicts that the total number of recalled items119872 at the end of the experiment is smaller than the numberof available items for recall This prediction was confirmedby the data Moreover the model-based analysis revealed astatistically significant positive correlation between 119872 and119862 for the participant sample studied in Frank and Rhodes[27]

35 Biology and Evolutionary Population Dynamics A pri-mary concern of evolutionary population dynamics is theunderstanding of the origin of life Among the many modelsthat have been proposed in this context the studies byCrutchfield and Gornerup [29] and Gornerup and Crutch-field [30] are of interest for our review In these studiesa number of 119873 macromolecules with different functional

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Stochastic AnalysisInternational Journal of

Page 7: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 7

processes In the case of processes defined on discrete statespaces 119875(119909 119905) is related to the occupation probability 119901(119896 119905)by means of (5) and (6) and (35) reduces to the special caseof (8) of the Markov chain model For processes defined oncontinuous state spaces and 120585 = 1 the general form (35)becomes (13) in the case of stochastic processes driven bynonparametric white noise In the limiting case 120585 rarr 0 weare dealing with time-continuous stochastic processes In thiscase the function 119891(sdot) possesses the property 119891(sdot) = 119883(119905)

for 120585 = 0 For processes that evolve on discrete state spacesand satisfy a nonlinear master equation it follows that (35)becomes the iterative map (22) provided that the argument119875(119909 119905) in 119891(sdot) is expressed in terms of the occupationprobability 119901(119896 119905) see (6) In contrast for processes evolvingon continuous state spaces and satisfying strongly nonlinearIto-Langevin equations the general form (35) reduces to thestochastic difference equation (26) when we consider thelimiting case 120585 rarr 0

3 Time-Discrete Strongly Nonlinear StochasticProcesses on Discrete State Spaces

31 The Strongly Nonlinear Two-State Markov Chain ModelA fundamental strongly nonlinear stochastic process is thetwo-state nonlinear Markov chain process [17] For the sakeof convenience the states 119896 = 1 and 119896 = 2will be labeled withldquo119860rdquo and ldquo119861rdquo Transitions from119860 to119860119860 to 119861 119861 to119860 and 119861 to119861 occur with conditional probabilities 119879(119860 rarr 119860) 119879(119860 rarr

119861) 119879(119861 rarr 119860) and 119879(119861 rarr 119861) respectively Accordinglythe nonlinear Markov chain equation (7) reads

119901 (119860 119905 + 1) = 119879 (119860 997888rarr 119860 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119905)) 119901 (119861 119905)

119901 (119861 119905 + 1) = 119879 (119860 997888rarr 119861 119905 119901 (119905)) 119901 (119860 119905)

+ 119879 (119861 997888rarr 119861 119905 119901 (119905)) 119901 (119861 119905)

(36)

Taking the normalization condition 119901(119860) + 119901(119861) = 1 intoconsideration (36) can equivalently be expressed by

119901 (119860 119905 + 1)

= [119879 (119860 997888rarr 119860 119905 119901 (119860 119905)) minus 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))] 119901 (119860 119905)

+ 119879 (119861 997888rarr 119860 119905 119901 (119860 119905))

(37)

which is a closed nonlinear equation in 119901(119860 119905) Model (37)involves two functions only the conditional probabilities119879(119860 rarr 119860) and 119879(119861 rarr 119860) which can be chosenindependently from each other and in the case of a nonlinearMarkov process depend on 119901(119860 119905) as indicated in (37)The remaining conditional probabilities can be calculatedfrom 119879(119860 rarr 119861) = 1 minus 119879(119860 rarr 119860) and 119879(119861 rarr

119861) = 1 minus 119879(119861 rarr 119860) Model (37) has been examined inphysics social sciences and psychology as will be discussednext

32 Chaos in Probability Time-Discrete Case FollowingFrank [18] let us put 119879(119860 rarr 119860) = 119879(119861 rarr 119860) = 119879

0 Then

(37) becomes

119901 (119860 119905 + 1) = 1198790(119905 119901 (119860 119905)) (38)

Although (38) describes a stochastic process formally (38)exhibits the structure of a nonlinear deterministic mapsubjected to the constraint that the function 119879

0must be in

the interval [0 1] at any time step 119905 and for any occupationprobability 119901(119860) Iterative maps in general are known toexhibit chaos [19] Therefore (38) suggests that time-discretestrongly nonlinear Markov processes exhibit chaos as wellSuch strongly nonlinear chaotic processes exhibit chaos in theevolution of the occupation probabilities and the conditionalprobabilities For example in Frank [18] the logistic function1198790(119911) = 120572 sdot 119911(1 minus 119911) [19 20] was considered for which (38)

reads

119901 (119860 119905 + 1) = 120572119901 (119860 119905) [1 minus 119901 (119860 119905)] (39)

For 120572 in [0 4] the function 1198790(sdot) is in the interval [0 1] such

that (39) maps the occupation probability 119901(119860) from theinterval [0 1] to the interval [0 1] as required For parameters120572 ge 120572crit asymp 3570 [20] the map exhibits chaotic solutionsThat is 119901(119860 119905) evolves in a chaotic fashion However asstated previously not only the occupation probability exhibitschaos but the conditional probabilities 119879(119860 rarr 119860) =

119879(119861 rarr 119860) = 1198790exhibit a chaotic behavior as well which

was demonstrated explicitly by numerical simulations [18]Chaos in strongly nonlinear time-discrete Markov processesis not limited to the two-state models In fact the two-statemodel can be generalized to an 119873-state model In this case(38) becomes [18]

119901 (119896 119905 + 1) = 1198790(119905 119901 (119905)) (40)

for 119896 = 1 119873 minus 1 The occupation probability 119901(119873 119905)

is computed from the normalization condition Conse-quently 119879

0depends only on the occupation probabilities

119901(1 119905) 119901(119873 minus 1 119905) For example the two-dimensionalchaotic Bakermap [20] was described bymeans of (40) using119873 = 3 [18]

33 Social Sciences andGroupDecisionMaking Time-DiscreteCase Group decision making has frequently been describedby nonlinear stochastic processes [21ndash24]The reason for thisis that under certain circumstances individuals are likely tobe affected by the opinion that is presented by a group asa whole rather than by the opinion of individuals This so-called group opinion in turn is produced by the opinions anddecisions of the individual group members Consequentlythe decision making process of an individual is affected bythe sample average obtained from the opinions of the othersIf the groups can be considered as being relatively large thesample averagemay be replaced by the ensemble averageThedecision making behavior of an individual is then consideredas a realization of a stochastic process that is affected by theensemble average of the process In the context of votingbehavior the two-state model introduced in Section 31 was

8 ISRNMathematical Physics

applied [17]Themembers of theUSHouse of Representativeshad the choice to vote for or against passing the so-calledbailout bill (US House of Representatives USA 2008) Thebill failed to pass in the September 29 vote In a second voteon October 3 the bill finally passedThat is a critical numberof ldquoNordquo voters changed their opinions between September 29and October 3 The states 119896 = 1 and 119896 = 2 were labeled byldquo119884rdquo and ldquo119873rdquo representing ldquoYesrdquo and ldquoNordquo voters Accordingto model (37) the conditional probabilities 119879(119873 rarr 119884) and119879(119884 rarr 119884) describing a change in the opinion from ldquoNordquo toldquoYesrdquo and a consistent ldquoYesrdquo opinion were considered Usingthe notation of (37) the voting model proposed in Frank [18]reads

119901 (119884 119905 + 1)

= [119879 (119884 997888rarr 119884 119905 119901 (119884 119905)) minus 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))] 119901 (119884 119905)

+ 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))

(41)

The data available in the literature suggested that 119879(119884 rarr

119884) = 1 For the119879(119873 rarr 119884) the function119879(119873 rarr 119884) = 119860minus119861sdot

119901(119884 119905) was used to describe the impact of the group opinionon the voting behavior This relationship assumes that thepressure to change from a ldquoNordquo voter to a ldquoYesrdquo voter was highimmediately subsequent to the September 29 votes when theprobability 119901(119884 119905) at 119905 = Sept 29 was not high enough tomakethe bill pass In this context it should be mentioned that thefailure of the bill to pass on September 29 triggered a dramaticbreakdown of the US stock market That is the members ofthe House of Representatives who had voted with ldquoNordquo werenot only subjected to political pressure but also experiencedessential pressure from the economy For 119879(119884 rarr 119884) = 1

and 119879(119873 rarr 119884) = 119860 minus 119861119901(119884 119905) the stationary occupationprobability 119901(119884 st) is given by 119901(119884 st) = 119860119861 such that (41)eventually reads

119901 (119884 119905 + 1) = 119901 (119884 119905) + 119861 [119901 (119884 st) minus 119901 (119884 119905)] [1 minus 119901 (119884 119905)]

(42)

The stationary value 119901(119884 st) was estimated from the dataA detailed model-based analysis of the data showed thatthe members of the two main parties the Democratic andRepublican members were differently affected by the grouppressure

34 Psychology and Human Memory A pioneering experi-ment in humanmemory research is the free recall experimentin which participants are asked to recall items of a particularcategory [25 26] For example they are asked to recall animalnames Typically participants are instructed to recall as manyitems as possible and to do so as fast as possible and withoutrepetitionThe total number of recalled items is by definitiona monotonically increasing function of time Strikinglyfree recall involves clusters in which item subcategories arerecalled more or less as a whole For example when recallinganimal names a participant may recall a number of insectnames in succession From a theoretical point of view at anygiven time point during the free recall process the ability

to recall items in clusters depends on the size of the poolof items available for recalling In other words if there isa large pool of items that have not yet been recalled thenthere is a high chance that a whole item subcategory canbe recalled In line with this argument a stochastic modelfor free recall dynamics was developed in which the recallprocess depends on the size of the pool of items availablefor recall [27] More precisely time was parceled in discretetime intervals in which recall attempts are executed The aimwas to develop a model for the probability that a single itemhas been or has not yet been recalled at a given time step 119905To this end the two-state model of Section 31 was appliedAccordingly the states 119896 = 1 and 119896 = 2 correspond tobeing recalled or being not yet been recalled and are labeledby ldquo119877rdquo and ldquo119873rdquo respectively The two relevant conditionalprobabilities are119879(119877 rarr 119877) and119879(119873 rarr 119877) By the design ofthe experiment we have119879(119877 rarr 119877) = 1 Moreover as arguedpreviously if the pool of items that have not yet recalled islarge that is if the probability 119901(119873) that an item has notyet been recalled is high then the conditional probability119879(119873 rarr 119877) should be high aswell Likewise if the probability119901(119877) that an item has been recalled is high then 119879(119873 rarr 119877)

should be relatively low Therefore it was suggested to put119879(119873 rarr 119877) = 119861 sdot (1 minus 119901(119877 119905)) with 119861 gt 0 This functionimplies that the occupation probability 119901(119877 119905) converges tothe stationary fixed 119901(119877 st) = 1 at which 119879(119873 rarr 119877) = 0The full model can be obtained by substituting 119879(119877 rarr 119877)

and 119879(119873 rarr 119877) into (37) and reads

119901 (119877 119905 + 1) = 119901 (119877 119905) + 119861[1 minus 119901 (119877 119905)]2

(43)

Note that from a mathematical point of view this modelis a special case of model (42) for voting behavior (hintput 119901(119884 st) = 1 in (42)) Since the model describes thesingle item recall probability the model must be scaled tothe number of items recalled by any given participant Tothis end a parameter 119862 describing the maximal number ofitems for recall was introduced The unknown parameters119861 and 119862 were estimated for free recall data collected in astudy by Rhodes and Turvey [28] and the sample meanvalues 119861 = 0009 (SD = 0005) and 119862 = 307 (SD = 138)were found for a sample of eight participants [27] Note that(43) predicts that the fixed point 119901(119877 st) = 1 will neverbe reached in a finite number of time steps Consequentlythe model predicts that the total number of recalled items119872 at the end of the experiment is smaller than the numberof available items for recall This prediction was confirmedby the data Moreover the model-based analysis revealed astatistically significant positive correlation between 119872 and119862 for the participant sample studied in Frank and Rhodes[27]

35 Biology and Evolutionary Population Dynamics A pri-mary concern of evolutionary population dynamics is theunderstanding of the origin of life Among the many modelsthat have been proposed in this context the studies byCrutchfield and Gornerup [29] and Gornerup and Crutch-field [30] are of interest for our review In these studiesa number of 119873 macromolecules with different functional

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

8 ISRNMathematical Physics

applied [17]Themembers of theUSHouse of Representativeshad the choice to vote for or against passing the so-calledbailout bill (US House of Representatives USA 2008) Thebill failed to pass in the September 29 vote In a second voteon October 3 the bill finally passedThat is a critical numberof ldquoNordquo voters changed their opinions between September 29and October 3 The states 119896 = 1 and 119896 = 2 were labeled byldquo119884rdquo and ldquo119873rdquo representing ldquoYesrdquo and ldquoNordquo voters Accordingto model (37) the conditional probabilities 119879(119873 rarr 119884) and119879(119884 rarr 119884) describing a change in the opinion from ldquoNordquo toldquoYesrdquo and a consistent ldquoYesrdquo opinion were considered Usingthe notation of (37) the voting model proposed in Frank [18]reads

119901 (119884 119905 + 1)

= [119879 (119884 997888rarr 119884 119905 119901 (119884 119905)) minus 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))] 119901 (119884 119905)

+ 119879 (119873 997888rarr 119884 119905 119901 (119884 119905))

(41)

The data available in the literature suggested that 119879(119884 rarr

119884) = 1 For the119879(119873 rarr 119884) the function119879(119873 rarr 119884) = 119860minus119861sdot

119901(119884 119905) was used to describe the impact of the group opinionon the voting behavior This relationship assumes that thepressure to change from a ldquoNordquo voter to a ldquoYesrdquo voter was highimmediately subsequent to the September 29 votes when theprobability 119901(119884 119905) at 119905 = Sept 29 was not high enough tomakethe bill pass In this context it should be mentioned that thefailure of the bill to pass on September 29 triggered a dramaticbreakdown of the US stock market That is the members ofthe House of Representatives who had voted with ldquoNordquo werenot only subjected to political pressure but also experiencedessential pressure from the economy For 119879(119884 rarr 119884) = 1

and 119879(119873 rarr 119884) = 119860 minus 119861119901(119884 119905) the stationary occupationprobability 119901(119884 st) is given by 119901(119884 st) = 119860119861 such that (41)eventually reads

119901 (119884 119905 + 1) = 119901 (119884 119905) + 119861 [119901 (119884 st) minus 119901 (119884 119905)] [1 minus 119901 (119884 119905)]

(42)

The stationary value 119901(119884 st) was estimated from the dataA detailed model-based analysis of the data showed thatthe members of the two main parties the Democratic andRepublican members were differently affected by the grouppressure

34 Psychology and Human Memory A pioneering experi-ment in humanmemory research is the free recall experimentin which participants are asked to recall items of a particularcategory [25 26] For example they are asked to recall animalnames Typically participants are instructed to recall as manyitems as possible and to do so as fast as possible and withoutrepetitionThe total number of recalled items is by definitiona monotonically increasing function of time Strikinglyfree recall involves clusters in which item subcategories arerecalled more or less as a whole For example when recallinganimal names a participant may recall a number of insectnames in succession From a theoretical point of view at anygiven time point during the free recall process the ability

to recall items in clusters depends on the size of the poolof items available for recalling In other words if there isa large pool of items that have not yet been recalled thenthere is a high chance that a whole item subcategory canbe recalled In line with this argument a stochastic modelfor free recall dynamics was developed in which the recallprocess depends on the size of the pool of items availablefor recall [27] More precisely time was parceled in discretetime intervals in which recall attempts are executed The aimwas to develop a model for the probability that a single itemhas been or has not yet been recalled at a given time step 119905To this end the two-state model of Section 31 was appliedAccordingly the states 119896 = 1 and 119896 = 2 correspond tobeing recalled or being not yet been recalled and are labeledby ldquo119877rdquo and ldquo119873rdquo respectively The two relevant conditionalprobabilities are119879(119877 rarr 119877) and119879(119873 rarr 119877) By the design ofthe experiment we have119879(119877 rarr 119877) = 1 Moreover as arguedpreviously if the pool of items that have not yet recalled islarge that is if the probability 119901(119873) that an item has notyet been recalled is high then the conditional probability119879(119873 rarr 119877) should be high aswell Likewise if the probability119901(119877) that an item has been recalled is high then 119879(119873 rarr 119877)

should be relatively low Therefore it was suggested to put119879(119873 rarr 119877) = 119861 sdot (1 minus 119901(119877 119905)) with 119861 gt 0 This functionimplies that the occupation probability 119901(119877 119905) converges tothe stationary fixed 119901(119877 st) = 1 at which 119879(119873 rarr 119877) = 0The full model can be obtained by substituting 119879(119877 rarr 119877)

and 119879(119873 rarr 119877) into (37) and reads

119901 (119877 119905 + 1) = 119901 (119877 119905) + 119861[1 minus 119901 (119877 119905)]2

(43)

Note that from a mathematical point of view this modelis a special case of model (42) for voting behavior (hintput 119901(119884 st) = 1 in (42)) Since the model describes thesingle item recall probability the model must be scaled tothe number of items recalled by any given participant Tothis end a parameter 119862 describing the maximal number ofitems for recall was introduced The unknown parameters119861 and 119862 were estimated for free recall data collected in astudy by Rhodes and Turvey [28] and the sample meanvalues 119861 = 0009 (SD = 0005) and 119862 = 307 (SD = 138)were found for a sample of eight participants [27] Note that(43) predicts that the fixed point 119901(119877 st) = 1 will neverbe reached in a finite number of time steps Consequentlythe model predicts that the total number of recalled items119872 at the end of the experiment is smaller than the numberof available items for recall This prediction was confirmedby the data Moreover the model-based analysis revealed astatistically significant positive correlation between 119872 and119862 for the participant sample studied in Frank and Rhodes[27]

35 Biology and Evolutionary Population Dynamics A pri-mary concern of evolutionary population dynamics is theunderstanding of the origin of life Among the many modelsthat have been proposed in this context the studies byCrutchfield and Gornerup [29] and Gornerup and Crutch-field [30] are of interest for our review In these studiesa number of 119873 macromolecules with different functional

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Stochastic AnalysisInternational Journal of

Page 9: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 9

properties were considered These macromolecules competewith each other for survival during evolutionary time scalesHowever not only competition but also cooperative behavioris taken into account Roughly speaking it is assumed that inthe early stages of the development of life macromoleculesof different types existed and could change their types dueto interactions with other macromolecules of the same ordifferent types Eventually only a subset of functionally dif-ferent types survived the selection process Let 119896 = 1 119873

denote the functionally different types of macromoleculesand 119901(119896 119905) the occupation probability of the 119896th type Thenat every discrete reaction time step 119905 the macromolecules canchange their types such that transitions from type 119894 to type 119896occur This so-called metamodel can be formulated in termsof a nonlinear Markov chain model [31] Accordingly theconditional (transition) probability 119879(119894 rarr 119896) describes theprobability of a transition from type 119894 to type 119896 given thata macromolecule is of type 119894 at time step 119905 As suggested bythe studies of Crutchfield and Gornerup [29] and Gornerupand Crutchfield [30] interactions between two different typesof macromolecules can be modeled by assuming that theevolution of occupation probabilities is affected by terms ofthe form 119901(119895 119905)119901(119894 119905) More precisely in general a macro-molecule of type 119894may interact with a macromolecule of type119895 and turn into a macromolecule of type 119896 It can be shownthat from the metamodel it follows that 119879(119894 rarr 119896) assumesthe form [31]

119879 (119894 997888rarr 119896) =

119873

sum

119895=1

119892 (119894119895119896) 119901 (119895 119905) (44)

In (44) the parameters 119892(119894119895119896) ge 0 are weight coefficients thatdescribe the relevance of interactions betweenmacromolecu-les of different types Note that the conditional probabilities119879(119894 rarr 119896) satisfy the normalization condition that the sumover all probabilities with index 119896 equals 1 This normaliza-tion can be guaranteed by requiring that the sum over allweight coefficients 119892 with index 119896 equals 1 Substituting (44)into (7) the metamodel for evolutionary population dynam-ics can be cast into a nonlinear Markov chain model andreads

119901 (119896 119905 + 1) =

119873

sum

119894119895=1

119892 (119894119895119896) 119901 (119895 119905) 119901 (119894 119905) (45)

It has been shown that the nonlinear Markov chain model(45) is a useful tool for investigating evolutionary populationdynamics First of all trajectories that describe the evolutionof individual macromolecules were computed numericallyusing (1) and (8)ndash(11) see Section 22 That is the modelallows for generating temporal sequences of type changesfor individual macromolecules Second it was shown thatthe degree to which a type 119896 is attractive can be quantifiedIn particular for evolutionary selection process that becomestationary the attractors of the surviving macromoleculetypes can be analyzed and distinctions between weakly andstrongly attractive types can be made (eg see Figure 3 in[31])

4 Time-Discrete Strongly Nonlinear StochasticProcesses on Continuous State Spaces

41 The Generalized Autoregressive Model of Order 1 as aStrongly Nonlinear Markov Process The generalized autore-gressive model of order 1 defined by (16) was studied inFrank [13] as time-discrete version of the time-continuousstrongly nonlinear Shimizu-Yamada model see Section 6Let us discuss this generalized autoregressive model in moredetail To make this discussion more self-consistent (16) ispresented here once again

119883(119905 + 1) = 119860119883 (119905) + 119861 ⟨119883 (119905)⟩ + 119892120576 (119905) (46)

as (46) Since the random variable 120576 exhibits a normal distri-bution at any time step 119905 the conditional probability densitycan be calculated that describes the distribution of 119883 at time119905 + 1 given the value of119883 = 119910 at the previous time step 119905 andthe probability density 119875(119909 119905) of 119883 at time step 119905 Assumingthat the variance of 120576 equals 2 the solution reads

119879 (119909 119905 + 1 | 119910 119905 120579119909[119875 (sdot 119905)])

= 119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩)

=1

119892radic4120587

exp(minus[119909 + 119860119910 + 119861 ⟨119883 (119905)⟩]

2

41198922

)

(47)

As indicated in (47) the conditional probability depends onlyon the first moments (ie the mean value) of the probabilitydensity 119875(119909 119905) By means of the conditional probability den-sity 119875(119909 119905 + 1) can be computed from 119875(119909 119905) like

119875 (119909 119905 + 1) = int

infin

minusinfin

119879 (119909 119905 + 1 | 119910 119905 ⟨119883 (119905)⟩) 119875 (119910 119905) 119889119910

(48)

From (46) it follows that the mean value 1198721(119905) = ⟨119883(119905)⟩

evolves like

1198721(119905 + 1)

= (119860 + 119861)1198721(119905) 997904rArr 119872

1(119905) = 119872

1(1) [119860 + 119861]

(119905minus1)

(49)

For 0 lt 119860 + 119861 lt 1 the mean value is an exponentiallydecaying function For minus1 lt 119860 + 119861 lt 0 the mean valuealternate between negative and positive values but decays inamplitude monotonically In summary for minus1 lt 119860 + 119861 lt 1

themean value converges to zero in the limiting case 119905 rarr infinA detailed calculation shows that the second moment 119872

2=

⟨1198832

(119905)⟩ evolves like

1198722(119905)

= 1198601198722(119905 minus 1) + 119860119861119872

1(119905 minus 1) + 119861119872

1(119905)1198721(119905 minus 1) + 2119892

2

(50)

For minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1 the first moment conver-ges to zero which implies that the second moment convergesto a stationary value given by 119872

2(st) = 2119892

2

(1 minus 1198602

) such

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 10: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

10 ISRNMathematical Physics

that the variance of the stationary processes is determinedby Var(119883) = 119872

2(st) Let us substitute the variable 119906 = 119892 sdot 120576

(which is a normally distributed random variable with vari-ance 21198922) into (40)Then for minus1 lt 119860+119861 lt 1 and minus1 lt 119860 lt 1

the stationary variance of119883 is given by

Var (119883) = Var (119906)1 minus 1198602 (51)

Not surprisingly (51) is just the expression for the varianceof the ordinary stationary autoregressive processes of order1 Moreover for normally distributed initial values 119883(1) theprobability density 119875(119909 119905) satisfies a normal distribution forall times 119905 This follows from (47) and (48) or alternativelyfrom the linearity of (46) Consequently for minus1 lt 119860 + 119861 lt 1

and minus1 lt 119860 lt 1 the probability density 119875(119909 119905) converges inthis case to a normal distribution with zero mean value andvariance given by (51) In the stationary case the correlationfunction Corr(119904) = ⟨119883(119905)119883(119905 minus 119904)⟩Var(119883) for 119904 ge 0 hasexactly the same form as the correlation function of theordinary autoregressive process of order 1 because the meanvalue119872

1equals zero and drops out of (46)

Corr (119904 + 1) = 119860 sdot Corr (119904) (52)

In conclusion for minus1 lt 119860 + 119861 lt 1 and minus1 lt 119860 lt 1

the generalized autoregressive process (46) behaves in thestationary case exactly as the ordinary autoregressive process(ie the process with 119861 = 0) The two processes howeverexhibit different transient characteristic properties

42 The Generalized Deterministic Autoregressive Process ofOrder 1 and the Sensitivity of Strongly Nonlinear Processes toInitial Conditions In the deterministic case we put 119892 = 0 in(13) such that

119883 (119905 + 1) = ℎ (119883 (119905) 119905 120579119883(119905)

[119875 (119909 119905)]) (53)

Let us consider the case 119873 = 1 and assume that the driftfunction ℎ(sdot) depends only linearly on the state 119883(119905) If inaddition ℎ(sdot) does not explicitly depend on time and 120579[sdot] is amap (functional or linear operator) that maps 119875(119909 119905) to a realnumber we arrive at the model

119883 (119905 + 1) = ℎ (120579 [119875 (119909 119905)]) 119883 (119905) = 119871 [119875 (119909 119905)]119883 (119905) (54)

In (54) we simplified the notation by introducing the operator119871(sdot) that maps 119875(119909 119905) to a real number (eg 119871 calculatesexpectation values of119883 and then uses them as arguments in afunction) Equation (54) may be considered as a generalizeddeterministic autoregressivemodel of order 1Model (54) wasstudied in detail in Frank [32] A striking feature of themodelis that the stationary probability density 119875(119909 st) if it existsdepends crucially on the initial probability density 119875(119909 119905 =

1) For sake of readability let us put 119906(119909) = 119875(119909 119905 = 1) Thenit can be shown that 119875(119909 st) if it exists is a rescaled functionof 119906(sdot) like [32]

119875 (119909 st) = 1

119911119906 (

119909

119911) (55)

with the scaling parameter

119911 = lim119904rarrinfin

119904

prod

119905=1

119871 [119875 (119909 119905)] (56)

In other words the strongly nonlinear process (54) mightexhibit infinitely many stationary probability densities Thiswas demonstrated more explicitly for a specific form of 119871(sdot)which will be discussed next

43 Economics and the Emergence of Stationary Volatilitiesas a Group Dynamics Process In economics the standarddeviation or the variance of a price of an asset is a measurefor how risky the assist is That is the variability measuresreflect the beliefs of traders how risky it is to hold the assetZero variability reflects a risk-free asset In this context thevariability of a price is also called the price volatility Volatilityis observed by traders and informs individual traders howrisky the market as a whole considers a particular asset Onthe other hand the market is composed of individual tradersAn autoregressive model of the form (54) can be used tomodel such a circular relationship To this end we assumethat the operator 119871(sdot) calculates the variance of119883 In this casethe evolution of 119883 is determined by the variance of 119883 onthe one hand On the other hand the variance by definitionis calculated from the realization of 119883 Let us consider thefollowing special case of (54) (for details see [32])

119883 (119905 + 1) = [1 minus 120574 (Var (119883) minus 120573)]119883 (119905) (57)

with 120574 120573 gt 0 It can be shown that for 120574 lt 2120573 the processexhibits stable but not asymptotically stationary probabilitydensitiesThe shape of these probability densities depends onthe initial distributions as discussed in Section 42 Howeverthe variance of all stable stationary distributions equals 120573This is intuitively clear when we realize that for Var(119883) = 120573(57) becomes the identity 119883(119905 + 1) = 119883(119905) Model (57) alsoexhibits unstable stationary probability densities One ofthese unstable solutions is the Dirac delta function 119875(119909) =

120575(119909) It can be shown that any process converges to a sta-tionary process with variance120573 provided that the initial prob-ability density of the process has a variance Var(119883) smallerthan a certain critical value For example if the Dirac deltafunction is perturbed slightly such that the variance becomesfinite but is still close to zero then the process will notconverge back to the Dirac delta function Rather the processwill converge to a stationary process with variance equal to 120573

We distinguished previously between stable and asymp-totically stable probability densities Stable but not asymptot-ically stable stationary solutions have also been called mar-ginally stable stationary solutions Recently research has beenfocused on the importance of marginally stabile stationarysolutions for biochemical and biological systems [33ndash35]In the context of strongly nonlinear processes stable butnot asymptotically stable probability densities or alternativelymarginally stable probability densities refer to processes thatexhibit a family of stationary probability densities with twoproperties [9] First by an infinitely small perturbation aparticular stationary probability density can be transformed

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 11

into another stationary probability density of the family ofprobability densities Second the family of probability densi-ties as awhole acts as an attractorThat is for any perturbationthat does not transform a member of the family into anothermember of the family the perturbed stationary probabilitydensity eventually converges to one of the members of thefamily of marginally stable probability densities A simpleexample can be given for model (57) As mentioned pre-viously for the marginally stable probability densities withVar(119883) = 120573 (57) reads 119883(119905 + 1) = 119883(119905) A shift of thecoordinate system is consistent with the identity 119883(119905 +

1) = 119883(119905) and does not affect the variance Therefore anystationary probability density with Var(119883) = 120573 can be shiftedalong the 119909-axis by an infinitely small amount to give rise toanother stationary probability density Clearly this kind ofperturbation will not decay

In closing the discussion on model (54) let us returnto the interpretation of the model as a model for volatilitydynamics The model predicts that the emergence of a sta-tionary volatility in the market is not guaranteed but dependson market parameters (here the inequality 120574 lt 2120573 mustbe satisfied) and the initial variance Furthermore the modelpredicts that the price itself is not affected whether or not astable stationary volatility measure emerges

44 Phase Synchronization of Inhomogeneous Ensemble ofGlobally Coupled Oscillatory Units Phase synchronizationof coupled oscillatory subunits has frequently been studiedby means of time-continuous strongly nonlinear stochasticmodel see Section 6 However also time-discrete modelshave been considered Phase synchronization is a specialcase of synchronization when the amplitude dynamics isneglected Accordingly each oscillatory unit is described by asingle real number representing a phase119883(119905) Let us envisioneach oscillatory unit as an oscillator evolving along a closedorbit in an appropriately defined state space Then the phase119883(119905) is the coordinate along that closed orbit The phase119883(119905)is a periodic variable The oscillators are desynchronized if119883 has a uniform distribution on the domain of definitionIf 119883 exhibits a nonuniform distribution (in particular aunimodal distribution) then the oscillators are partiallysynchronized If 119883 = 119860 for all oscillators then there iscomplete synchronization

Let us consider an all-to-all coupling between the oscilla-tors In this case the evolution of the phase119883(119905) of an individ-ual oscillator depends on the evolution of all other oscillatorphases If the set of oscillators is relatively large the limitof infinitely many oscillators may be considered as a goodapproximation and the sample of oscillators may be replacedby a statistical ensemble In this case the phase dynamicsof an individual oscillator depends on the expectation valueof the statistical ensemble of oscillators Moreover any indi-vidual oscillator represents a realization of the ensemble Insummary a closed description of the phase dynamics in termsof a strongly nonlinear stochastic process can be obtained

In the deterministic case we put 119892 = 0 in (13) such that(13) becomes (53) mentioned in Section 42 Without loss ofgenerality the period can be put equal to unity such that119883 is

defined in the interval [0 1] and 119883 = 0 and 119883 = 1 indicatethe same location on the closed orbit In line with seminalworks by Kuramoto [36] Daido [37] proposed an explicitform of (53) that can be written for individual realizationslike

119883119903

(119905 + 1) = 119883119903

(119905) + 119880119903

minus 120581119902

2120587sin (2120587119883119903 (119905))

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (2120587119883119903 (119905)) (58)

The variable 119902 is the order parameter of the oscillator systemFor desynchronized oscillators (ie for a uniformly dis-tributed119883)we have 119902 = 0 If 119902differs fromzero the oscillatorsare partially synchronized In (58) the parameter 120581 ge 0

describes the coupling strength between the oscillators Moreprecisely it is the coupling between individual oscillators andthe mean field force defined by ℎMF(119883) = 119902 sdot sin(2120587119883)(2120587)that acts on an individual oscillator and is generated by alloscillators In (58) the parameter119880 denotes the phase velocityfor the uncoupled oscillator As indicated 119880 is taken froma distribution and differs among the oscillators Thereforemodel (58) describes an inhomogeneous ensemble of globallycoupled oscillatory units

Model (58) is a useful model for studying the emer-gence of phase synchronization from the perspective ofnonequilibrium phase transitions At a critical value of thecoupling parameter 120581 a nonequilibrium phase transitionoccurs and the probability density 119875(119909) bifurcates from auniform distribution to a nonuniform distribution indicatingthat the ensemble makes a transition from a desynchronizedstate to a partially synchronized state [37] For Lorentziandistributed phase velocities 119880 an analytical expression ofthe critical coupling parameter can be found [37] Similaraspects of time-discrete models for synchronizations havebeen studied inDaido [38 39] and by Teramae andKuramoto[40]

Following more recent work [32] the conditional prob-ability density of the phase synchronization model (58) interms of the so-called Frobenius-Perron operator involvingthe Dirac delta function 120575(sdot) can be determined and reads

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

= 120575 (mod [119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) 1])

119902 = int

1

0

cos (2120587119909) 119875 (119909 119905) 119889119909

(59)

The modulus-1 operator may be eliminated by casting (59)into the form

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

=

infin

sum

119895=minusinfin

120575 (119909 minus 119910 minus 119880 + 120581119902

2120587sin (2120587119910) + 119895)

(60)

The conditional probability density holds for a particularunit with phase velocity 119880 Let 120588(119880) describe the probability

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

12 ISRNMathematical Physics

density of phase velocities Then 119875(119909 119905 + 1) can at leastformally be computed from 119875(119909 119905) and 120588(119880) via the integralequation

119875 (119909 119905 + 1)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot 119905)] 119880)

times 119875 (119910 119905) 120588 (119880) 119889119910 119889119880

(61)

Accordingly stationary solutions 119875(119909 st) satisfy

119875 (119909 st)

= int

0infin

119910=1119880=minusinfin

119879 (119909 119905 + 1 | 119883 (119905) = 119910 120579 [119875 (sdot st)] 119880)

times 119875 (119910 st) 120588 (119880) 119889119910 119889119880(62)

The uniform distribution 119875(119909) = 1 satisfies (62) for anycoupling parameter 120581 ge 0 and consequently is a stationaryprobability density However for oscillator systems withLorentzian velocity distributions 120588(119880) the uniform distri-bution is an unstable stationary probability density if thecoupling parameter exceeds the critical value which can beshown by numerical simulations [37]

5 Time-Continuous StronglyNonlinear Stochastic Processes onDiscrete State Spaces

Strongly nonlinear stochastic processes evolving on discretestate spaces but exhibiting a continuous time variable havebeen studied in the context of nonlinear master equations asintroduced in Section 24 that is in the context of Markovprocesses Frequently nonlinear master equations have beenstudied as a tool to derive nonlinear Fokker-Planck equations(see eg [14 41ndash44]) However nonlinear master equationshave also been studied in their own merit (see eg [45])

51 Nonlinear Master Equations as a Tool for Identifyingthe Generic Forms of Nonlinear Fokker-Planck Equations Asmentioned in Section 24 transition rates ofmaster equationsmay depend on occupation probabilities Curado and Nobre[14] considered only transition rates between neighboringstates 119896 In addition the explicit time dependency of rates wasneglected In this case (20) reads

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 997888rarr 119896 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 997888rarr 119896 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 997888rarr 119896 + 1 119901 (119905)) + 119908 (119896 997888rarr 119896 minus 1 119901 (119905))]

(63)

Note that there are only two types of transition rates Firstthere are transition rates that describe the rate of transitionsfrom a level to the next higher level (ldquoupward ratesrdquo) Secondthere are the transition rates of transitions from a level to thenext lower level (ldquodown ratesrdquo) Using 119908(119896 up 119901) = 119908(119896 rarr

119896 + 1 119901) and 119908(119896 down 119901) = 119908(119896 rarr 119896 minus 1 119901) we can re-write (63) as

119889

119889119905119901 (119896 119905)

= 119908 (119896 minus 1 up 119901 (119905)) 119901 (119896 minus 1 119905)

+ 119908 (119896 + 1 down 119901 (119905)) 119901 (119896 + 1 119905)

minus 119901 (119896 119905) [119908 (119896 up 119901 (119905)) + 119908 (119896 down 119901 (119905))] (64)

Furthermore Curado and Nobre [14] assumed that the states119896 represent positions on a one-dimensional grid with spacingΔ They assumed that the transition rates depend on thespacing Δ but also on the occupation probabilities 119901(119896 119905)More precisely the model involves the following up- anddownrates

119908 (119896 up 119901 (119905)) = 119860

Δ2[119901 (119896 119905)]

119902minus1

119908 (119896 down 119901 (119905)) = minus1

Δℎ (119896Δ) +

119860

Δ2[119901 (119896 119905)]

119902minus1

(65)

In the limiting case of an infinitely small grid (ie forΔ rarr 0) the master equation (64) with (65) behaves likethe nonlinear Fokker-Planck equation of the form (29) Thediffusion term of that Fokker-Planck equation is proportionalto the probability density119875119902Thismodel in turn is a standardmodel for diffusion in porous media [46 47] see Section 6

52 Strongly Discrete-Valued Nonlinear Stochastic ProcessesMaximizing Generalized Entropy Measures A key propertyof stochastic processes described by ordinary master equa-tions such as (20) with rates 119908(119894 rarr 119896) independent of theoccupation probabilities 119901(119896) is that the processes approachstationarity in the long time limit [48] More preciselyprocesses evolve such that the Kullback distance measure[49] is a monotonically decreasing function of time and ap-proaches a stationary value The stationarity of the Kullbackdistance measure indicates that the process under considera-tion has become stationary The Kullback distance measureis defined on the basis of the Boltzmann-Gibbs-Shannonentropy which is a fundamental entropy measure not onlyfor equilibrium systems [50] but also for nonequilibriumsystems [51] It was suggested to exploit this link between theBoltzmann-Gibbs-Shannon entropy the Kullback distancemeasure and the ordinary master equation to define non-linear master equations based on generalized entropy andgeneralized Kullback distance measures [9 45] Accordinglyentropy measures 119878 of the form

119878 (119901) =

119873

sum

119896=1

119904 (119901 (119896)) (66)

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Page 13: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 13

are considered that involve a kernel function 119904(sdot) Further-more it is assumed that the kernel 119904(sdot) satisfies certainproperties The second derivative of the kernel 119904(119911) withrespect to 119911 is negative for any argument 119911 in the interval[0 1] which implies that the entropy 119878(119901) is concave [52] Inaddition the first derivative of the kernel 119904(119911) with respectto 119911 in the limiting case 119911 rarr 0 goes to plus infinity (ieexhibits a singularity) This property is important for themaster equation that will be defined in the following andguarantees that occupation probabilities 119901(119896) solving themaster equation do not become negative The followingmaster equation has been proposed [45]

119889

119889119905119901 (119896 119905)

=

119873

sum

119894=1

119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)] minus 119865 [119901 (119896 119905)]

119873

sum

119894=1

119860 (119896 997888rarr 119894 119905)

119865 (119911) = expminus1 minus 119889119904 (119911)

119889119911

(67)

The nonlinearity in the master equation (67) is conceptuallydifferent from the nonlinearity in the master equation (20)While in (20) it is assumed that the transition rates depend onoccupation probabilities in (67) the transition rates 119860(119894 rarr

119896) are independent of the occupation probabilities Howevertransitions are not proportional to the occupation probabil-ities 119901(119896) as in (20) Rather they are proportional to theterms 119865(119901(119896)) whose explicit form depends on the entropykernel 119904(sdot) In view of this property transitions are entropydriven It can be shown that stochastic processes satisfying(67) converge to occupation probabilities that maximize theentropymeasures 119878Therefore wemay say that the transitionsof processes defined by (67) are proportional to an entropydrive 119865(sdot) that maximizes the entropy 119878 under considerationNote that for the special case of the Boltzmann-Gibbs-Shannon entropy the function 119865 reduces to the identity119865(119911) = 119911 which implies that (67) includes the ordinary linearmaster equation as special case

In closing our considerations on the nonlinear masterequation (67) it is useful to note that formally (67) can becast into the form of the nonlinear master equation (20)irrespective of any conceptual differences If we put

119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) 119901 (119894 119905) = 119860 (119894 997888rarr 119896 119905) 119865 [119901 (119894 119905)]

997904rArr 119908 (119894 997888rarr 119896 119905 119901 (119894 119905)) = 119860 (119894 997888rarr 119896 119905)119865 [119901 (119894 119905)]

119901 (119894 119905)

(68)

for119901(119894 119905) gt 0 thenmodel (67) assumes the formof (20)Notethat in the limiting case 119901 rarr 0 we have 119865 rarr 0 because ofthe aforementioned assumption that 119889119904(119911)119889119911 rarr infin holdsfor 119911 rarr 0 This in turn implies that 119908(119894 rarr 119896 119905 119901(119894 119905))

for 119901(119894 119905) = 0 can be defined as the product of 119860(119894 rarr

119896 119905) and the limiting case ldquo00rdquo where the ratio ldquo00rdquo has toevaluated for any given entropy kernel 119904(119911) The relation (68)

is useful because it can be used to compute realizations of thenonlinear master equation (67) from (1) (18) and (22)ndash(25)

53 Physiological Partitioned (Discrete-Valued) Signals andStrongly Nonlinear Stochastic Processes Exhibiting StationaryCut-OffDistributions It has been argued that any probabilitydensity with tails that extend to infinity fails to capture theboundedness of physiological signals [53] In other wordswhile probability densities with tails that extend to infinitysuch as the normal distribution are crucially importantfor developing theory and for modeling and are frequentlygood approximations they imply that signals can assumearbitrarily large values in magnitude This is not plausiblePhysiological signals such as muscle force produced byhuman and animals limb velocities limb accelerations bloodflow velocities heart rates electroencephalographic recordedbrain signals pulse rates of neurons and dendritic currentsare bounded and cannot exceed certain maximum valuesTaking an information theoretical point of view [51] stochas-tic processes describing physiological signals may maximizeinformation or entropy measures However they cannotmaximize the Boltzmann-Gibbs-Shannon measure underordinary constraints because this measure yields normaldistributions or other distributions with tails that extendto infinity In contrast physiological signals may maximizegeneralized information and entropy measures that exhibitthe so-called cut-off distributions as maximum entropydistributions This argument was developed for stochasticprocesses defined on continuous state spaces [53] but holdsfor processes evolving on discrete-state spaces as well Inparticular signals from sensory systems are known to bepartitioned in a certain way Differences between two magni-tudes can only be detected if they exceed certain thresholdsFor this reason modeling of sensory physiological signalsand physiological signals in general may assume that the statespace of consideration is of discrete nature

In Frank [45] the coordination of rhythmicmovements oftwo limbs has been addressed exploiting the master equationmodel (67) It has been assumed that the physiological rele-vant variable namely the relative phase between the limbsis appropriately described on a partitioned (ie discrete-value) state space Furthermore it has been assumed that thecoordination dynamics corresponds to a stochastic processmaximizing an entropy measure that exhibits a maximumentropy cut-off distribution Out of all possible entropy mea-sures the entropy measure nowadays called Tsallis entropyhas been used for that purpose [54ndash56] The model can beconsidered as a modified version of the seminal coordinationmodel by Haken et al [57] The benefits of the nonlinearmaster equation model (67) relative to the original Haken-Kelso-Bunz model are twofold First model (67) offers aconsistent approach that addresses physiological data withinthe framework of cut-off distributions Second the modelpredicts that coordination is bistable in a distributional senseThat is under certain conditions the model exhibits twostationary distributions 119901(119896 st) This feature is in line withexperimental observations For human coordination modelswith continuous state spaces similar attempts have beenmadeto deal with bistability in a distributional sense [58 59]

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Stochastic AnalysisInternational Journal of

Page 14: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

14 ISRNMathematical Physics

6 Time-Continuous StronglyNonlinear Stochastic Processes onContinuous State Spaces

There is a relatively large literature on time-continuousstrongly nonlinear stochastic processes defined on continu-ous state spaces Most of these studies are concerned with theMarkov diffusion processes Reviews can be found in Frank[9 60] Strongly nonlinear phase synchronization modelsbased on the Kuramoto model [36] are reviewed in Acebronet al [61] In this section some of applications in physics andthe life sciences will be highlighted without going into muchdetail

61 Chaos inProbabilityTime-ContinuousCase In Section 32strongly nonlinear time-discrete stochastic processes wereconsidered that exhibit occupation probabilities that evolvein a chaotic fashion As an example the logistic map modeldefined by (39) was discussed A key feature of thismodel andof similar models is that the chaotic behavior arises from thecoupling of the dynamics of the realizations to the occupationprobabilities In particular without this coupling equation(39) reads 119901(119860 119905 + 1) = 119879

0= 1205724 with fixed parameters

120572 taken from the interval [0 4] and clearly does not exhibita chaotic solution Chaos is due to the fact that transitionprobability 119879

0= 1205724 is replaced by a transition probability

that depends on the process occupation probabilities like1198790=

120572sdot119901(119860 119905)(1minus119901(119860 119905))That is probabilitymeasures of stronglynonlinear processes can exhibit chaotic behavior due to thevery nature of strongly nonlinear processes which is theinfluence of a probabilitymeasure on realizations fromwhichthe probabilitymeasure is calculated Such a circular causalitystructuremay be realized inmany-body systems composed ofinteracting subsystems In such systems chaos may emergedue to the coupling of the single subsystem dynamics to therelative frequency with which states are occupied by subsys-tems Chaos emerges even if the isolated subsystem dynamicsis not chaotic [18] Consequently chaos may be considered asa self-organization phenomenon Recall that self-organizingphenomena in general are emerging properties ofmany-bodysystems that do not exist on the level of the subsystems [62]In our context the many-body system as a whole behavesqualitatively differently from the individual isolated partsThe catchphrases ldquothe whole is different from the sum of itspartsrdquo or ldquomore is differentrdquo apply [63]

This feature of strongly nonlinear stochastic processes hasalso been demonstrated for time-continuous models involv-ing continuous state spaces [64ndash67] Subsystems described byordinary stochastic differential equations that do not exhibitchaotic solutions have been coupled together and the limitingcase of a many-body system composed of infinitely manysubsystems has been considered The limiting case modelswere described by strongly nonlinear stochastic processesin the sense defined in Section 12 In particular in thesestudies the dynamics of realization was coupled to certainexpectation values of the respective processes It was foundthat the expectation values under appropriate conditionsexhibit a chaotic dynamics

62 Plasma Physics and the Landau Form of Strongly Non-linear Fokker-Planck Equations Plasma physics is concernedwith the evolution of many-particle systems composed ofcharged particles The particles can interact with each otherin various ways Two interaction forms are particle-particlecollisions and Coulomb interaction The Landau form of theFokker-Planck equation accounts for these two interactionsand describes the evolution of the particle density in thethree-dimensional velocity space That is let V = (V

1 V2 V3)

denote the velocity vector of a particle Let 120588(V 119905) denotethe particle mass density at time 119905 Assuming that particlesare neither created nor destroyed the total mass 119872 isinvariant over time The particle probability density 119875(V 119905) =120588(V 119905)119872 can be introduced According to the Landau formthe probability density 119875(V 119905) satisfies a strongly nonlinearFokker-Planck equation of the form (29) More precisely theLandau form reads [68ndash74]

120597

120597119905119875 (V 119905) = minus

3

sum

119896=1

120597

120597V119896

[ℎ119896(V 120579V [119875 (sdot 119905)]) 119875 (V 119905)]

+

3

sum

119895=1119896=1

1205972

120597V119895120597V119896

[119863119895119896(sdot) 119875 (V 119905)]

ℎ119896(V 120579V [119875 (sdot 119905)]) = 119860

120597

120597V119896

int

Ω

119875 (V1015840

119905)

1003816100381610038161003816V minus V1015840

1003816100381610038161003816

1198893

V1015840

119863119895119896(sdot) = 119861

1205972

120597V119895120597V119896

int

Ω

10038161003816100381610038161003816V minus V101584010038161003816100381610038161003816119875 (V1015840

119905) 1198893

V1015840

(69)

Stationary solutions 119875(V st) of (69) have been studied forexample by Soler et al [75] In addition several studies havebeen concerned with the derivation of transient solutions119875(V 119905) using different (semi)numerical approaches [74 76ndash79]

63 Astrophysics Stellar Cut-Off Mass Distributions Definedby Strongly Nonlinear Stochastic Models Astrophysical massdistributions may exhibit relative sharp boundaries Moreprecisely particle distributions in the stellar space may bespatially bounded due to gravity The gravitational forces areproduced by the mass distributions themselves That is thedynamics of an individual particle of a mass distribution isaffected by the particle distribution as a whole In turn thedistribution is composed of its individual particles In viewof this circular causality it does not come as a surprise thatstrongly nonlinear stochastic models have been investigatedthat describe cut-off distributions of stellar particles [80ndash84] The models typically assume the form of the stronglynonlinear Fokker-Planck equation (29)

Two more comments may be appropriate First inSection 53 we addressed cut-off distributions in the con-text of physiological signals However the motivation inSection 53 for considering cut-off distributions was quitedifferent from the argument used in astrophysical applica-tions Second in addition to the aforementioned studies onstrongly nonlinear processes describing astrophysical cut-off mass distribution strongly nonlinear stochastic processes

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Page 15: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 15

satisfying the Landau form (69) have also been used forstudying astrophysical problems [85 86]

64 Accelerator Physics Shortening of Bunch-Particle Distri-bution of Particle Beams Particles in accelerator beams cantravel in localized cluster of particles In a longitudinal beamthe so-called bunch-particle distribution is a mass densitythat describes the distribution of particles with respect tothe center of the cluster Let 119909

1denote relative position of

a particle with respect to the bunch center Typically theparticle dynamics also depends on the particle energy Thatis beam particle dynamics involves a second coordinate 119909

2

which is a rescaled energy measure (rather than a velocityor momentum variable) Dividing the mass density withthe total mass the bunch-particle distribution becomes aprobability density 119875(119909 119905) of the two-dimensional vector 119909 =

(1199091 1199092) Particles in accelerator beams are charged particles

that are accelerated by means of electromagnetic forcesAlthough particles may interact with each other by means ofCoulomb forces the crucial force that determines the shapeand length of a cluster is the so-called wake-field [87 88]The wake-field is an electromagnetic field generated by thewhole bunch of particles that travels ahead of the bunch butis reflected at the accelerator walls and then acts back on theparticles of the bunch The wake-field can cause a shorteningof the particle bunch That is under the impact of the wake-field the cluster is smaller in size than it would be theoreticallyin the absence of the wake-fieldThe understanding of bunchshortening is an important step towards an understanding ofbeam instabilities that should be avoided

The dynamics of bunch particles is typically described bystrongly nonlinear Markov diffusion processes The reasonfor this is that the dynamics of individual particles is affectedby thewake-fieldwhich can be calculated froman appropriateexpectation value of the bunch particle probability density119875(119909 119905) As a result in various studies it has been proposed that119875(119909 119905) satisfies a strongly nonlinear Fokker-Planck equationof the form (29) (see eg [89ndash98]) A useful model in partic-ular for the purpose of theory development is the Haissinskimodel [95 96 99ndash104] for which analytical solutions of thereduced density119875(119909

1) can be derived In particular according

to the Haissinski model the dynamics of single particlessatisfies a strongly nonlinear Langevin equation (27) whichreads [100]

119889

1198891199051198831(119905) = 119883

2(119905)

119889

1198891199051198832(119905) = ℎ (119883 (119905) 120579

1198831(119905)[119875 (119909 119905)]) + 119892Γ

2(119905)

ℎ = minus 1198831(119905) minus 120574119883

2(119905)

minus120597

1205971198831

int

Ω

119880119882(1198831minus 1199091015840

1) 119875 (119909

1015840

1 1199092 119905) 119889119909

1015840

11198891199092

(70)

where 120574 gt 0 describes a phenomenological dampingconstant For the Haissinski model the wake-field function119880119882

assumes the form of a Dirac delta function 119880119882(119911) =

120581 sdot 120575(119911) The parameter 120581measures the strength of the impact

of the wake-field For 120581 lt 0 the width of the reducedprobability density 119875(119909

1) becomes smaller when increasing

the magnitude |120581| of the impact of the wake-field In doingso model (70) provides a dynamic account for the squeezingeffect of wake-fields on particle clusters traveling in accelera-tor beams

65 Physics of Liquid Crystal Phase Transitions from thePerspective of Strongly Nonlinear Stochastic Processes Liquidcrystals typically consist of rod-shape macromolecules thatinteract with each other by means of weak Van-der-Waalsforces Due to this interaction molecules can align them-selves such that the molecules point with their primary axesmore or less in the same direction The physical propertiesof a crystal with aligned molecules are qualitatively differentfrom the properties that the crystal exhibits when the macro-molecules are isotropically (randomly) distributed There-fore liquid crystals exhibit two phases an ordered phase withmolecules that are aligned at least to some degree which iscalled the nematic phase and a disorder phasewithmoleculespointing in random directions which is called the isotropephase A phase transition from the isotropic to the nematicphase occurs when the temperature is decreased below acritical temperature [105ndash108] The degree of orientationalorder can be captured by the Maier-Saupe order parameter[102 105ndash111] For the isotropic phase the order parameterequals zero In the nematic phase the Maier-Saupe orderparameter is larger than zero

Stochastic models describing the emergence of orienta-tional order (ie isotropic-nematic phase transitions of liquidcrystals) exploit the notion that the order parameter itselfexpresses the magnitude of an overall force that acts onindividual molecules and tends to align them with the meanorientation of the liquid crystal macromolecules The modelis based conceptually on the notions of circular causality andself-organization individual molecules are affected by theorder parameter and at the same time constitute the orderparameter Hess [112] as well as Doi and Edwards [113] haveproposed a strongly nonlinear stochastic process describingthe rotational Brownian motion of the molecules under theimpact of the (self-generated) order parameter The Hess-Doi-Edwards model exhibits the structure of a strongly non-linear Fokker-Planck equation (29) and can be cast into theform of a strongly nonlinear Langevin equation (27) (see eg[114]) Exploiting the concept of strongly nonlinear stochasticprocesses the martingale for the Hess-Doi-Edwards modelcan be defined [60]

Transient solutions of the Hess-Doi-Edwards model maybe computed from approximate coupled moment equations[115] Using Lyapunovrsquos direct method [62 116] it can beshown that the ordered state exhibits an asymptotically stableprobability density [114] Importantly since stochasticmodelsprovide a dynamic account it is possible to study responsefunctions of liquid crystals Transient responses describingthe relaxation to equilibrium [117 118] as well as correlationfunctions and mean square displacement functions in thestationary case [114] can be determined at least semi-numer-ically

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Page 16: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

16 ISRNMathematical Physics

Beyond the application of the concept of strongly non-linear stochastic processes to the Hess-Doi-Edwards Fokker-Planck equation in general strongly nonlinear stochasticmodels have turned out to be usefulmodels in liquid polymerphysics (see eg [119ndash121])

66 Classical Descriptions of Quantum Systems by meansof Strongly Nonlinear Stochastic Processes The relaxationtowards equilibrium of quantummechanical Fermi and Bosesystems can be modeled using a classical approach by meansgeneralized Boltzmann equations that exhibit appropriatelymodified collision integrals [122ndash125] Applying a diffusionapproximation to these collision integrals such quantummechanical Boltzmann equations reduce to Fokker-Planckequations that are nonlinear with respect to their probabilitydensities [122ndash124] In addition to this approach via the Boltz-mann equation alternative derivations of classical quantummechanical Fokker-Planck equations that are nonlinear withrespect to probability densities have been proposed [126ndash133] A key property of these models is that they exhibitstationary probability densities that correspond to Fermi orBose distributions Interpreting these models in terms ofstrongly nonlinear Fokker-Planck equations of form (29)allows us not only to consider the evolution of probabilitydensities but also to examine predictions of semiclassicalstochastic processes for quantum systems For examplesolutions of trajectories computed from the correspondingstrongly nonlinear Langevin equations of form (27) have beenstudied [126 127] In particular the relaxation of the freeelectron gas towards its stationary Fermi energy distributioncan be determined by computing numerically individualelectron trajectories in the relevant energy space [9 126]Moreover martingales for quantum processes within thisclassical Langevin approach can be defined [60]

Finally note that classical stochastic descriptions of Fermiand Bose systems do not necessarily involve strongly non-linear stochastic processes It has been shown that ordinaryFokker-Planck equations with appropriately defined driftfunctions ℎ(sdot) can also exhibit Fermi and Bose distributionsas stationary probability densities (see eg [134])

67Material Physics of PorousMedia Gas particles and fluidsdiffusing through porous media satisfy a nonlinear diffusionequation frequently called the porous media equation Inone spatial dimension the porous medium equation for theparticle mass density 120588(119909 119905) reads [3 9 46 47 135]

120597

120597119905120588 (119909 119905) = 119860

1205972

1205971199092[120588 (119909 119905)]

119902

(71)

with 119860 gt 0 Dividing the mass density 120588(119909 119905) by the totalmass119872 (71) becomes an evolution equation for a probabilitydensity 119875(119909 119905) normalized to unity

120597

120597119905119875 (119909 119905) = 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(72)

with 119863 = 119860 sdot 119872(119902minus1) and assumes the form of the nonlinear

Fokker-Planck equation (29) The connection between theporous medium equation and nonlinear master equations

has been addressed in Section 51 (see the aforementioned)Equations (71) and (72) exhibit exact time-dependent solu-tions frequently called Barenblatt-Pattle solutions [136 137]

The parameter 119902 captures the nonlinearity of the equa-tion For 119902 = 1 (linear case) the porous medium equationreduces to the ordinary diffusion equations and the varianceof the diffusion process (or the second moment of 119875(119909 119905)assuming a zero first moment) increases linearly with time119905 In contrast for 119902 gt 13 and 119902 = 1 (nonlinear case) thevariance increases as a nonlinear function of 119905 like 119905

119903 with119903 = 2(1+119902) as can be shownby variousmethods [9 138ndash140]

Interpreting (72) in terms of a strongly nonlinear Fokker-Planck equation (29) trajectories of realizations can be com-puted from the corresponding strongly nonlinear Langevinequation [138] For a generalized version of (72) involvinga drift force such a Langevin equation approach has beenexploited to derive analytical expressions of certain autocor-relation functions [141]

The porous medium equation in its form as a nonlinearFokker-Planck equation (72) plays an important role in thetheory development of nonextensive thermostatistics As itturns out a particular generalization of (72) the so-calledPlastino-Plastino model exhibits stationary solutions thatmaximize the nonextensive entropy proposed by Tsallis Wewill return to this issue later

68 Material Physics and the Liouville Dynamics of Many-Body Systems with Interacting Subsystems The particle dis-tribution of a many-body system in the overdamped deter-ministic case satisfies a Liouville equation For many-bodysystems with interacting subsystems the Liouville equationinvolves an integral term describing the interaction force on asingle subsystem This interaction integral may be evaluatedusing a diffusion approximation In doing so the Liouvilleequation assumes the form of the nonlinear Fokker-Planckequation (29)when considering the probability density ratherthan the subsystem density The nonlinearity occurs in thediffusion term This approach has been developed in aseries of studies by Andrade et al [142] and Ribeiro etal [143 144] For example the distribution functions ofinteracting particles in dusty plasmas interacting vorticesand interacting polymer chains in complex fluids have beenstudied within this framework

Note that as mentioned previously the dynamics of thesubsystems is deterministic Nevertheless the effectivemodelfor the subsystem probability density involves a diffusiontermThat is according to the effective approximative modelin terms of a nonlinear Fokker-Planck equation due to theinteraction between the subsystems the many-body systemas a whole generates a fluctuating force that acts on theindividual subsystems of the many-body system

69 Nonextensive Physical Systems and the Plastino-PlastinoModel Tsallis (Tsallis [54 55] see also Abe and Okamoto[56]) introduced in physics a generalized form of the Boltz-mann-Gibbs-Shannon entropy nowadays frequently calledthe Tsallis entropy The Tsallis entropy exhibits a parameter119902 For 119902 = 1 the entropy reduces to the Boltzmann-Gibbs-Shannon entropy capturing properties of extensive systems

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Page 17: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 17

For 119902 = 1 the Tsallis entropy describes nonextensive systemsA R Plastino and A Plastino [145] proposed a nonlinearFokker-Planck equation of the form (29) that may be writtenlike

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909) 119875 (119909 119905)] + 119863

1205972

1205971199092[119875 (119909 119905)]

119902

(73)

The model describes the probability density 119875(119909 119905) of aparticle living in a one-dimensional continuous state spacegiven by the whole real line The term involving a forcefunction ℎ(119909) in (73) is linear with respect to the probabilitydensity 119875(119909 119905) The diffusion term of the model involves theparameter 119902 which using the notation suggested by (73)corresponds to the parameter 119902 of the Tsallis entropy (seeFrank [9]) Note that in the original model a slightly differentnotation was proposed [145] A key feature of the Plastino-Plastino model (73) is that stationary probability densities119875(119909 st) of (73)maximize the Tsallis entropy For 119902 = 1 that isin the special case in which the Tsallis entropy reduces to theBoltzmann-Gibbs-Shannon entropy the model is linear withrespect to the probability density 119875(119909 119905) For 119902 = 1 that is forthe general case that the Tsallis entropy describes a nonexten-sive system the diffusion term is nonlinear with respect to119875(119909 119905) In view of these observations the Plastino-Plastinomodel establishes a connection between nonextensivity andnonlinearity

The Plastino-Plastino model (73) has been examined invarious studies In particular it has frequently been pointedout that (73) may be considered as a generalized version ofthe porous medium model (72) for gas and fluid flows thatare subjected to an additional driving force captured by theforce function ℎ(119909)

Exact analytical solutions for the time-dependent proba-bility density 119875(119909 119905) are available in the case of a linear driftforce [140 145 146] Trajectories describing the stochasticdynamics of individual particles can be computed wheninterpreting (73) as a strongly nonlinear Fokker-Planckequation by means of the corresponding strongly nonlinearLangevin equation [138 141 147] In this case second orderstatistics measures such as autocorrelation functions can bedetermined [141] and martingales can be found [60] Exacttime-dependent solutions have been derived for generalizedversions of model (73) that account for source terms [148ndash150] The Kramers escape rate has been determined forprocesses described by the Plastino-Plastinomodel involvingan appropriately defined drift force [151] The multivariategeneralization of (73) with [139 152 153] and without [129154] time-dependent coefficients has been considered ThePlastino-Plastino model (73) with explicit state-dependentdiffusion coefficients 119863(119909) has been studied [153 155] Themodel has been generalized for the Sharma-Mittal entropythat involves two parameters and includes the Tsallis entropyas a special case [156] Interestingly H-theorems have beenfound that show that transient solutions 119875(119909 119905) of (73) con-verge to stationary probabilities densities 119875(119909 st) providedthat ℎ(119909) is chosen appropriately [122 157ndash164]

610 Oscillatory Coupled Nonequilibrium Systems Canonical-Dissipative Approach Coupled oscillators with short-range

interactions can be studied within the framework of stronglynonlinear stochastic processes [165] In this study isolatedoscillators were modeled using the canonical-dissipativeapproach [166ndash168] that allows to exploit the concepts ofHamiltonian andNewtonian dynamics on the one hand andto address nonequilibrium systems such as self-excited limitcycle oscillators on the other hand

611 Phase Synchronization and Kuramoto Model Time-Continuous Case Synchronization is a phenomenon studiedin various disciplines ranging from physics to the socialsciences [169] In the context of strongly nonlinear stochas-tic processes we are primarily concerned with the syn-chronization of a large ensemble of oscillatory subunitsSynchronization can be studied both in the phase spaceof the oscillators (eg the space spanned by position andmomentum variables) and in reduced spaces An examplefor the latter case was discussed already in Section 44 phasesynchronization In fact we will focus in this section on thephenomenon of phase synchronization

A benchmark model that describes the emergence ofphase synchronization in a population of phase oscillatorsis the Kuramoto model [36] Accordingly each oscillator isdescribed by a phase 119883 that evolves on a continuous timescale 119905 and represents a periodic variable Each oscillator iscoupled to all remaining oscillators and the limiting caseof a group of infinitely many oscillators is considered Theoscillators as a whole generate an order parameter whichis a complex-valued variable The complex-valued orderparameter has an amplitude the so-called cluster amplitude119860 and an angle the so-called cluster phase 120572 Both clusterphase 120572 and cluster amplitude 119860 are measures defined incircular statistics (see eg [36 170 171]) Mathematicallyspeaking for an oscillator phase 119883 defined on the interval[0 2120587] the complex order parameter 119902 is defined by theexpectation value

119902 (120579 [119875 (119909 119905)]) = lim119877rarrinfin

1

119877

119877

sum

119903=1

exp (119894119883119903 (119905))

= int

2120587

119909=0

exp (119894119909) 119875 (119909 119905) 119889119909

= 119860 (119905) exp (119894120572 (119905))

(74)

where 119894 is the imaginary unit (ie the square root of minus1) andthe cluster phase 120572 is chosen such that119860 ge 0 in any case Notethat both 119860 and 120572 are time-dependent variables

The order parameter 119902 induces a force that acts on theindividual phase oscillators That is the oscillators interactwith each other indirectly via the self-generated order param-eter The population of phase oscillators can be regardedas a statistical ensemble Accordingly the order parametercan be computed as an expectation value from the proba-bility density 119875(119909 119905) that is the probability density of thephase of an arbitrarily chosen phase oscillator Since theorder parameter acts back on the realizations (individualphase oscillators) the model satisfies the definition of astrongly nonlinear stochastic process provided in Section 12

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Page 18: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

18 ISRNMathematical Physics

The strongly nonlinear Langevin equation of the Kuramotomodel reads

119889

119889119905119883 (119905) = minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ (119905)

(75)

and assumes the form (27) In (75) the parameter 120581 ge 0

measures the strength of the force induced by the orderparameter 119902

The uniform probability density 119875 = 1(2120587) describing adesynchronized oscillator population is a stationary solutionfor any parameter 120581 because for 119875 = 1(2120587) the clusteramplitude 119860 equals zero However only for 120581 lt 2119892

2 theuniform distribution is asymptotically stable For 120581 gt 2119892

2 theuniform distribution is unstable meaning that for any initialcondition that slightly deviates from the uniform probabilitydensity 119875 = 1(2120587) the strongly nonlinear stochastic processwill not converge to a stationary process with uniformprobability density Rather for 120581 gt 2119892

2 the Kuramotomodel exhibits stable stationary unimodal distributions [936] These unimodal probability densities indicate that theoscillator population is partially synchronized Moreover theorder parameter amplitude 119860 is larger than zero for theseunimodal stationary probability densities The unimodalprobability densities are stable but not asymptotically stable(see eg [9]) A shift of such a stationary probability densityalong the 119909-axis transforms a member of the family of stablestationary probability densities into another member of thatfamily This feature becomes intuitively clear when we realizethat the form of (75) is invariant against transformation119883 rarr 119883 + 119904 and 120572 rarr 120572 + 119904 where 119904 is an arbitrarilysmall real number We may use the term ldquomarginallyrdquo stableto characterize the stability of these stationary probabilitydensities (see also Section 43)

The Kuramoto model has been reviewed in Strogatz[172] and Acebron et al [61] In particular there exists anH-theorem of the Kuramoto model showing that transientprobability densities converge to stationary ones [173] ThisH-theorem is related to a free energy approach to theKuramoto model (Frank [174] see also Frank [9])

A key question in the context of the Kuramoto modelconcerns the bifurcation from the desynchronized state to thesynchronized state for oscillator populations with inhomoge-neous eigenfrequencies In this case (75) may be written forrealizations119883119903 of the process like

119889

119889119905119883119903

(119905)

= 119880119903

minus 120581119860 (120579 [119875 (119909 119905)]) sin (119883119903 (119905) minus 120572 (120579 [119875 (119909 119905)]))

+ 119892Γ119903

(119905)

(76)

where 119880119903 is the phase velocity of the 119903th realization of

the process If we restrict our considerations to symmetricdistributions of velocities and to symmetric initial probabilitydensities then the symmetry property remains conserved

during the evolution of the process and the cluster phase canassume only one of the two values 120572 = 0 or 120572 = 120587 Moreoverthe order parameter 119902 becomes a real-valued variable It canbe shown that in this case (76) reduces to

119889

119889119905119883119903

(119905) = 119880119903

minus 120581119902 sin (119883119903 (119905)) + 119892Γ119903

(119905)

119902 = lim119877rarrinfin

1

119877

119877

sum

119903=1

cos (119883119903 (119905)) (77)

Comparing (77) with the Daido model (58) discussed inSection 44 it becomes at this stage clear that the Daidomodel (58) can be considered as a time-discrete counterpartof the time-continuous fully symmetric Kuramoto model(77) with distributed eigenfrequencies Note that the Daidomodel exhibits phases defined on the interval [0 1] whereasin the context of theKuramotomodel typically phases definedon the interval [0 2120587] are considered

As mentioned previously reviews for the Kuramotomodel are available in the literature and the reader mayconsult these works for more details ([61 172] see also Tass[175] and Section 75 in [9])

612 Biology Population Dispersion and Population Dynam-ics The spatial distribution of insects forming swarms maybe described in terms of cut-off distributions For examplethe insect density 120588(119909) = 119860 sdot [1 minus 119861 sdot |119909|

+]2 has been

proposed [176] where the coordinate xmeasures the locationof an insect with respect to the center of the swarm and119860 119861 gt 0 The operator sdot

+corresponds to the identify for

positive arguments and equals zero for negative arguments(ie 119911

+= 119911 for 119911 gt 0 and 119911

+= 0 otherwise) Normalizing

120588 by the total mass119872 a stochastic model for the probabilitydensity 119875(119909) = 120588(119909)119872 needs to explain the emergence ofa stationary insect cut-off probability density In fact to thisend a model similar to the Plastino-Plastino model (73) withan appropriate drift term was proposed [176 177]

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[120574 sgn (119909) 119875 (119909 119905)]

+ 119863120597

120597119909[119875 (119909 119905)]

120583120597

120597119909119875 (119909 119905)

(78)

with 120574 gt 0 The stationary probability 119875(119909 st) = 119862 sdot [1minus

119861 sdot |119909|+]2 is obtained for 120583 = 05 However exponents

different from 120583 = 05 have also been proposed [178 179]Moreover descriptions of population dynamics in terms ofnonlinear Fokker-Planck equations alternative to (78) havebeen considered in the literature as well ([81 168] see alsothe Shigesada-Teramoto model in Okubo and Levin [176]Section 56)

613 Social Sciences and Group Decision Making Time-Continuous Case Group decision making has already beenaddressed in Sections 33 and 43 in the context of time-discrete models In a series of studies Boster and coworkersdeveloped the so-called linear discrepancy model of group

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 19: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 19

decision making and compared predictions with experimen-tal data [180ndash182] In particular the model accounts for akey phenomenon in the social sciences the risky shift Arisky shift occurs when people arrive at riskier decisionswhen discussing a given topic in a group than when makingtheir own decisions individually According to the lineardiscrepancy model due to discussions the opinion in favoror against a particular issue shifts by an amount that isproportional to the opinion that an individual holds and theopinion that is presented in the group by another individualThe linear discrepancymodel predicts that a risky shift can beobserved when the initial distribution (ie the prediscussiondistribution) of opinions is skewed Accordingly during thediscussion session the opinion distribution tends to becomemore symmetric which is accompanied with a shift of themean value Someof themathematical properties of the lineardiscrepancy model have been studied in more detail withinthe framework of a strongly nonlinear stochastic process[183] Interestingly the stationary symmetric probabilitydensities describing the distribution of opinions among thegroup members are only stable and not asymptotically stableIn particular a shift along the opinion axis transforms onemember of the family of stable stationary probability densityinto another member In this regard the group decisionmaking model exhibits the same feature as the Kuramotomodel

614 Finance and Option Pricing A stochastic option pricingmodel has been developed on the basis of the Plastino-Plastino model (73) In particular the option pricing modelexhibits a diffusion term similar to the one of the Plastino-Plastino model [184ndash186] A particular benefit of the modelis that it features non-Gaussian distributions This is due tothe nonlinearity of the diffusion term (or the nonextensivityof the Tsallis entropy measure involved in the definition ofthe process) In fact non-Gaussian distributions frequentlyfit financial data better than Gaussian distributions

615 Economics andModeling theDynamics of Target LeverageRatios The total capital (firm value) of a company is typicallycomposed of borrowed money and equity The leverage of acompany or the leverage ratio is the ratio of borrowedmoneyrelative to equity A company needs to decide what leverageratio the company should haveThedecision is not necessarilymade once and for all Rather the desired leverage ratio(ie the target leverage ratio) may be updated dynamicallydepending on the internal and external circumstances Forthis reason the leverage ratios of companies frequentlycorrespond to dynamic variables subjected to fluctuations

Lo andHui [187] proposed a strongly nonlinear stochasticmodel for the dynamics of the leverage ratio The modelis based on a model developed earlier by Hui et al [188]that involved an explicitly time-dependent function In thestrongly nonlinear stochastic model this function is elim-inated and in this sense the strongly nonlinear stochasticmodel can be regarded as being closed In order to achievethis closure Lo and Hui [187] assumed that the leveragedynamics of a company is affected by the leverage ratios

of similar companies Within the framework of stronglynonlinear stochastic processes this implies that the leverageratio dynamics of companies depends on an expectationvalue of the leverage ratio process Formally the model by Loand Hui is closely related to the Shimizu-Yamada model thatwill be discussed later

616 Engineering Sciences Generators for Noise Sources Inengineering sciences and related disciplines a common prob-lem is to simulate numerically signals of noise sourcesThese signals can then be used to test the response ofengineered systems to random signals emerging from noisesources A characteristic feature of noise sources is that theyare stationary In view of this property strongly nonlinearstochastic processes can be defined which for any initialprobability density are stationary [147 189 190] For examplethe strongly nonlinear Langevin equation

119889

119889119905119883 (119905) = minus119860119883 (119905) + 119892 (119883 (119905) 120579 [119875 (119909 119905)]) Γ (119905)

119892 = radic119861

119875(119910 119905)[119862 minus int

119910

minusinfin

119875(119911 119905)119889119911]

10038161003816100381610038161003816100381610038161003816119910=119883(119905)

(79)

with 119860 119861 119862 gt 0 corresponds to a nonlinear Fokker-Planck equation of the form (29) whose right-hand side is bydefinition equal to zero [9 147] Consequently the Langevinequation defines for any initial probability density 119875(119909 119905 =

0) a stochastic process with a stationary probability density119875(119909) The parameters119860 119861 119862 can be chosen in order to selecta desired correlation time of the process

617 Further Benchmark Models

6171 The Shimizu-Yamada Model The Shimizu-Yamadamodel is motivated by research on muscle contraction(Shimizu [191] and Shimizu and Yamada [192] see also Sec-tion 554 in [9])The Shimizu-Yamadamodel corresponds tothe generalized Ornstein-Uhlenbeck process that is definedby (33) and involves a mean-field force proportional to themean value of the process The Shimizu-Yamada model ishighly relevant for the theory development of strongly non-linear Markov diffusion processes because it can be solvedexactly That is exact transient solutions can be found andexact solutions for the conditional probability density 119879(119909 119905 |119910 119904 ⟨119883(119904)⟩) are availableThe conditional probability densityin turn can be used to calculate all correlation functionsof interest both in the transient and in the stationary casesFor details see Frank [9 193] Since the model exhibitsexact solutions it has also been used to test the accuracy ofnumerical algorithms for solving nonlinearMarkov diffusionprocesses [16]

6172 The Desai-Zwanzig Model The Desai-Zwanzig modelinvolves a mean-field force proportional to the mean valueof the process just as the Shimizu-Yamada model Howeverthe individual isolated subsystems are assumed to performan overdamped motion in a double-well potential [194 195]

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Stochastic AnalysisInternational Journal of

Page 20: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

20 ISRNMathematical Physics

The strongly nonlinear Fokker-Planck equation of the Desai-Zwanzig model reads

120597

120597119905119875 (119909 119905) = minus

120597

120597119909[ℎ (119909 120579 [119875 (119909 119905)]) 119875 (119909 119905)] + 119863

1205972

1205971199092119875 (119909 119905)

ℎ (119909 120579 [119875 (119909 119905)]) = 119860119909 minus 1198611199093

minus 120581 (119909 minus119872 (119905))

119872 (119905) = int

infin

minusinfin

119909119875 (119909 119905) 119889119909

(80)

for a stochastic process defined on the whole real line and119860 119861 120581 gt 0 The term 119860119909 minus 119861119909

3in (80) describes thepotential force derived from the aforementioned double-wellpotentialThe term minus120581(119909minus119872) describes the mean-field forceproportional to the mean value of the process The forceattracts the realizations of the process towards themean valueof the process The parameter 120581 measures the strength ofthe mean-field force The model is symmetric with respectto 119909 = 0 In view of this observation it is intuitively clearthat the model exhibits a symmetric stationary probabilitydensity with 119872 = 0 for all parameters 120581 It can be shownthat for parameters 120581 smaller than a certain critical valuethis symmetric probability density is asymptotically stableand is the only stationary probability density Howeverwhen 120581 exceeds the critical value the symmetric stationaryprobability density becomes unstable Two asymptoticallystable stationary probability densities emerge with meanvalues 119872 = 119911 and 119872 = minus119911 where 119911 gt 0 [9 194 196]The mean value M has typically been considered as orderparameter of a many-body system described by the stronglynonlinear stochastic process (80) For 119872 = 0 the many-body system exhibits a disordered state For119872 = 0 the systemexhibits some order Therefore the Desai-Zwanzig modeldescribes an order-disorder phase transition

The special importance of the Desai-Zwanzig model isits feature that provides a fundamental stochastic modelfor an order-disorder phase transition The Desai-Zwanzigmodel and the Kuramoto model may be viewed as the twokey models in this regard Both models are able to captureorder-disorder phase transitionsWhile the Kuramotomodelis a prototype system for many-body systems with statevariables subjected to periodic boundary conditions theDesai-Zwanzig model is a prototype system for many-bodysystems featuring state variables satisfying natural boundaryconditions Moreover the Desai-Zwanzig model has played acrucial role in the development of theH-theorem for stronglynonlinear Fokker-Planck equations (we will return to thisissue later)

Various studies have addressed at least to some extentthe Desai-Zwanzig model This can be shown by interpretingthe Desai-Zwanzig model in the context of the free energyapproach to nonlinear Fokker-Planck equations [9] Accord-ingly the Desai-Zwanzig model does not only involve themean value M of the process but also the process variance119870 = ⟨119883

2

⟩minus1198722 In order to see this we need to take advantage

of variational calculus Let 120575119870120575119875 denote the variational

derivative of 119870 with respect to the probability density 119875(119909)A detailed calculation shows that

119872 = int

infin

minusinfin

119909119875 (119909) 119889119909 119870 = int

infin

minusinfin

1199092

119875 (119909) 119889119909 minus1198722

997904rArr120575119870

120575119875= 1199092

minus 2119909119872

997904rArr1

2

120597

120597119909

120575119870

120575119875= 119909 minus119872

(81)

Exploiting this result the free energy 119865 can be defined like

119865 [119875] = int

infin

minusinfin

119881 (119909) 119875 (119909) 119889119909 +120581

2119870 [119875] minus 119863119878 [119875]

119878 [119875] = minusint

infin

minusinfin

119875 (119909) ln (119875 (119909)) 119889119909

(82)

where 119878 denotes the Boltzmann-Gibbs-Shannon entropy ofthe probability density 119875(119909) The potential 119881(119909) denotes thedouble-well potential 119881(119909) = minus119860119909

2

2 + 1198611199094

4 The stronglynonlinear Fokker-Planck equation (80) can then equivalentlybe expressed by [9 194 196]

120597

120597119905119875 (119909 119905) =

120597

120597119909[119875 (119909 119905)

120597

120597119909

120575119865

120575119875] (83)

where 120575119865120575119875 is the variational derivative of the free energy119865 with respect to the probability density 119875(119909 119905) Accordingto (82) and (83) the Desai-Zwanzig model involves thevariance 119870 in the free energy It has been suggested to callstrongly nonlinear stochastic processes ldquovariance modelsrdquoor ldquo119870 modelsrdquo provided that they involve the variationalderivatives of the variance (ie they involve a coupling tothe process mean value) A minireview of the Desai-Zwanzigmodels and related ldquovariance modelsrdquo or ldquo119870 modelsrdquo can befound in Frank [9] Finally note that the Shimizu-Yamadamodel discussed in Section 6171 that is the generalizedOrnstein-Uhlenbeck model (33) belongs to the class ofldquovariance modelsrdquo as well

618 Shiinorsquos H-Theorem for Nonlinear Fokker-Planck Equa-tions As mentioned in Section 6172 the Desai-Zwanzigmodel played a crucial role in the theory development of theH-theorem for strongly nonlinear Fokker-Planck equationsWhile it was well known that stochastic processes describedby ordinary Fokker-Planck equations under appropriate cir-cumstances converge to stationary processes in the long timelimit the situation in this regard was not clear for stronglynonlinear Markov diffusion processes described by stronglynonlinear Fokker-Planck equations In physics the proofof convergence to the stationary case is typically given interms of a so-called H-theorem In a seminal work Shiino[196] provided an H-theorem for the Desai-Zwanzig modelThis H-theorem was generalized and discussed in variousstudies [122 157ndash164 173 197ndash201] see also our comments in

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Page 21: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 21

Section 68 and 610 for H-theorems related to the Plastino-Plastino model and the Kuramoto model respectively Aselection of H-theorems can be found in Frank [9] Finallynote that one can show that not only the single time pointprobability density 119875(119909 119905) of a strongly nonlinear stochasticprocess converges to a stationary probability density But thewhole process becomes stationary as well [202]

In the context of Shiinorsquos work on the H-theorem wewould like to point out that the study by Shiino [196] alsoestablished a novel approach to conduct a stability analysisfor nonlinear Fokker-Planck equation A minireview of thestability analysis by Shiino can be found in Frank [9]

7 Mean Field Theory and Strongly NonlinearStochastic Processes

While from a mathematical point of view strongly nonlinearstochastic processes are well defined the question arises towhat extent strongly nonlinear stochastic processes describephysical systems In other words we may ask what kind ofphysical systems give rise to strongly nonlinear stochasticprocesses An important class of physical systems that hasbeen discussed in this context is the class of many-bodysystems that can be treated at least approximately with thehelp of mean-field theory (see eg [36 175 195 203 204])The departure point is a many-body system composed of 119877subsystems The subsystems are coupled with each other bymeans of an all-to-all coupling which involves an averageover a function119882 of the state variables of the subsystemsThelimiting case 119877 rarr infin (the so-called thermodynamic limit)is considered next In this limiting case it is assumed that(i) the subsystems become statistically independent of eachother and (ii) the average over 119882 becomes an expectationvalue of119882 of an arbitrarily chosen representative subsystemThe function 119882 frequently represents a component of aforce or corresponds to a force itself This force is called amean-field force A key problem in this regard is to providea mathematical rigorous proof that in the limiting case119877 rarr infin the many-body system composed of 119877 subsystemscan be regarded as a statistical ensemble of realizationsThat is it is often a challenge to prove the convergence ofan ordinary multivariate stochastic process to a stronglynonlinear stochastic process In particular the mathematicalliterature is concerned with this problem (see the following)

An alternative bottom-up approach to strongly nonlinearstochastic processes uses as departure point a many-bodysystem in the limiting case 119877 rarr infin Again the subsystemsare assumed to be coupled by means of an average over afunction119882 of the subsystem state variables Furthermore itis assumed that the subsystems of the many-body system areinitially statistically independent and identically distributedIt can then be shown with relatively little effort that ifan arbitrary finite subset of size 119871 is considered then thesubsystems of this subset remain statistically independent atall times The implication is that in the limiting case 119871 rarr

infin the subset converges to a statistical ensemble and themany-body system can be described by means of a stronglynonlinear stochastic process [9 202]

8 Strongly Nonlinear Stochastic Processes inthe Mathematical Literature Mean-FieldApproach H-Theorems and Martingales

In the mathematical literature nonlinear Markov processeshave been discussed in the monograph by Kolokoltsov [205]Besides the seminal work byMcKean that opened the avenueto the novel research field on strongly nonlinear stochasticprocesses (see Sections 11 and 12) the mathematical litera-ture on strongly nonlinear stochastic processes has tradition-ally been concerned with the convergence of a multivariatestochastic process to a strongly nonlinear stochastic processin the limiting case in which the system size goes to infinity[7 195 206ndash210] That is the argument advocated by mean-field theory presented in Section 7 has been examined inthose studies from amathematical perspective A second lineof inquiry concerns the convergence of processes towardsstationarity That is the issue of the existence of H-theoremsdiscussed in Section 617 has also attracted the attentionof researchers in mathematics [197 211ndash214] Furthermorethe existence of martingales for strongly nonlinear Markovprocesses has been pointed out in the mathematical literature[7 208 209 215ndash220] and has been taken from there tophysics and the life sciences [60]

Strongly nonlinear stochastic processes have also beenapproached from two perspectives slightly different from themean-field theory approach Dawson et al [221] approachedstrongly nonlinear stochastic processes from the perspectiveofmeasure-valued processes whereas Stroock [222] provideda geometrical approach to the notion of strongly nonlinearMarkov processes

Finally nonlinear Fokker-Planck equations have fre-quently been addressed in the mathematical literature inthe context of semilinear and nonlinear parabolic partialdifferential equations In this context the reader may consultthe books by Friedman [223 224] and Logan [225] and theworks by Milstein and colleagues [226ndash228] on numericalsolution methods for nonlinear parabolic partial differentialequations

9 Strongly NonlinearNon-Markovian Processes

The examples reviewed in the previous sections belong tothe class of Markov processes The definition of stronglynonlinear stochastic processes given in Section 12 howeverdoes not imply that strongly nonlinear stochastic processessatisfy the Markov property In fact non-Markovian stronglynonlinear stochastic processes can be definedwith little effortFor example while the generalized autoregressive model oforder 1 defined by (16) satisfies the Markov property thegeneralized autoregressive model of order 119901 defined by

119883(119905) =

119901

sum

119896=1

(119860119896119883 (119905 minus 119896) + 119861

119896119872(119905 minus 119896)) + 119892120576 (119905)

119872 (119905) = lim119877rarrinfin

1

119877

119877

sum

119903=1

119883119903

(119905)

(84)

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Page 22: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

22 ISRNMathematical Physics

with119860119901

= 0 or 119861119901

= 0 describes a strongly nonlinear stochas-tic non-Markovian process for 119901 gt 1 Likewise stronglynonlinear stochastic non-Markovian processes have beenstudied evolving on continuous time scales and continuousstate spaces [229ndash231] in particular in the context of phasesynchronization and the Kuramoto model [232] and inthe context of muscle contraction and the Shimizu-Yamadamodel [233]

10 Coda

Let us highlight some aspects of strongly nonlinear stochasticprocesses

Any closed description of an evolution equation for therealization of a process provides a complete and well-defineddescription of the process because all quantities of interest canat least numerically be computed from (an infinitely larger)ensemble of realizations Therefore in Section 12 stronglynonlinear stochastic processes have been defined on the basisof evolution equations for realizations of stochastic processesIn Section 2 closed descriptions of such evolution equationshave been provided In view of these considerations weconclude that strongly nonlinear stochastic processes arewell defined That is it is advocated here that stronglynonlinear stochastic processes are processes worth beingstudied in their ownmerit irrespective of underlying physicalmodels or derivations based for example on generalizedthermostatistics (Section 68) or mean-field theory (Sections7 and 8)

The term ldquostrongly nonlinearrdquo should be considered as aplaceholder for a variety of other phrases that have been usedin the literature to label processes of the kind discussed inthis paper In particular it should be considered as a synonymto the phrase ldquotruly nonlinearrdquo Using the phrase ldquostronglynonlinearrdquo we may also regard ldquoweakly nonlinearrdquo processesSuch processes exhibit stochastic evolution equations forrealizations that do not depend on expectation values orprobability densities but feature other types of nonlineari-ties for example nonlinear functions of the relevant statevariables The stochastic Gompertz model and the stochasticlogistic model [234] of population growth are two examplesof processes that may be called ldquoweakly nonlinearrdquo

From our considerations in Sections 2 to 6 it becomesclear that the concept of strongly nonlinear stochastic pro-cesses is a fundamental one and does not depend on theexplicit representation of time and space In particularstrongly nonlinear stochastic processes are not restricted tothe class of processes originally considered by McKean Thatis strongly nonlinear stochastic processes are not limited tothe class of diffusion processes evolving on continuous statespaces and on a continuous time scale

Likewise as it obvious from our discussion in Section 9the concept of strongly nonlinear stochastic processes and theconcept of Markov processes are two independent conceptsA strongly nonlinear stochastic process may or may notexhibit the Markov property In this context it should bementioned that textbook definitions of the Markov propertyfrequently give rise to some confusion because such textbookdefinitions typically make use of the notion of a ldquostate

variablerdquo However the key issue concerning the Markovproperty is the clear separation between future presence andpast Accordingly a process is a Markov process if the futuredepends only on the present characteristic properties of theprocess and is independent of the characteristic propertiesthat the process exhibited in the past Subsequently we mayask the question about what are the characteristic propertiesThese properties include the state variables of the process butmay also include the occupation probabilities for a processdefined on discrete states or the probability density for aprocess defined on a continuous state space That is in thecase of a Markov process the present state as well as thepresent probability density determines entirely the futureof a realization of a process The knowledge about statesprior to the present state as well as the knowledge aboutthe probability density at times prior to the present timeis irrelevant This notion is at the heart of the definitionof a Markov process [13 60] and applies to the processesreviewed in Sections 2 to 6 In this context the fact thatin the mathematical literature Markov diffusion processeshave been frequently related to martingales (see Section 8)should be highlighted because martingales nicely illustratethe dependency of the future of a process on the presentproperties of the process and the independency of the futureof the process on its past properties

References

[1] H P McKean Jr ldquoA class of Markov processes associatedwith nonlinear parabolic equationsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 56 pp1907ndash1911 1966

[2] H P McKean Jr ldquoPropagation of chaos for a class of nonlinearparabolic equationsrdquo in Lectures in Differential Equations A KAziz Ed vol 2 pp 177ndash193 Van Nostrand Reinhold CompanyNew York NY USA 1969

[3] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[4] C W Gardiner Handbook of Stochastic Methods SpringerBerlin Germany 1985

[5] S Karlin and H M Taylor A first course in stochastic processesAcademic Press New York NY USA 1975

[6] H RiskenTheFokker-Planck EquationMethods of Solution andApplications Springer Berlin Germany 1989

[7] S Meleard ldquoAsymptotic behaviour of some interacting particlesystems McKean-Vlasov and Boltzmann modelsrdquo in Proba-bilistic Models for Nonlinear Partial Differential Equations CGraham T G Kurtz S Meleard P E Potter M Pulvirenti andD Talay Eds vol 1627 pp 42ndash95 Springer Berlin Germany1996

[8] M FWehner andW GWolfer ldquoNumerical evaluation of path-integral solutions to Fokker-Planck equations III Time andfunctionally dependent coefficientsrdquo Physical Review A vol 35no 4 pp 1795ndash1801 1987

[9] T D Frank Nonlinear Fokker-Planck Equations SpringerBerlin Germany 2005

[10] P J Diggle Time Series A Biostatistical Introduction TheClarendon Press Oxford UK 1990

[11] M B Priestley Nonlinear and Nonstationary Time Series Anal-ysis Academic Press London UK 1988

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 23: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 23

[12] T D Frank ldquoMarkov chains of nonlinear Markov processesand an application to a winner-takes-all model for socialconformityrdquo Journal of Physics A vol 41 no 28 Article ID282001 2008

[13] T D Frank ldquoNonlinear Markov processesrdquo Physics Letters Avol 372 no 25 pp 4553ndash4555 2008

[14] E M F Curado and F D Nobre ldquoDerivation of nonlinearFokker-Planck equations by means of approximations to themaster equationrdquo Physical Review E vol 67 no 2 Article ID021107 2003

[15] T D Frank ldquoStochastic feedback nonlinear families of Markovprocesses and nonlinear Fokker-Planck equationsrdquo Physica Avol 331 no 3-4 pp 391ndash408 2004

[16] D S Zhang G W Wei D J Kouri and D K HoffmanldquoNumerical method for the nonlinear Fokker-Planck equationrdquoPhysical Review E vol 62 no 1 pp 3167ndash3172 1997

[17] T D Frank ldquoDeterministic and stochastic components of non-linear Markov models with an application to decision makingduring the bailout votes 2008 (USA)rdquo European Physical JournalB vol 70 no 2 pp 249ndash255 2009

[18] T D Frank ldquoChaos from nonlinear Markov processes why thewhole is different from the sum of its partsrdquo Physica A vol 388no 19 pp 4241ndash4247 2009

[19] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[20] H G Schuster Deterministic Chaos An Introduction VCHVerlagsgesellschaft Weinheim Germany 1988

[21] S Galam and J-D Zucker ldquoFrom individual choice to groupdecision-makingrdquo Physica A vol 287 no 3-4 pp 644ndash6592000

[22] B Latane and A Nowak ldquoMeasuring emergent social phenom-ena dynamism polarization and clustering as order parame-ters of social systemsrdquo Behavioral Science vol 39 pp 1ndash24 1994

[23] F Schweitzer and J A Hołyst ldquoModelling collective opinionformation by means of active Brownian particlesrdquo EuropeanPhysical Journal B vol 15 no 4 pp 723ndash732 2000

[24] M Takatsuji ldquoAn information theoretical approach to a systemof interacting elementsrdquo Biological Cybernetics vol 17 no 4 pp207ndash210 1975

[25] W A Bousfield and C H Sedgewick ldquoAn analysis of restrictedassociative responsesrdquo Journal of General Psychology vol 30 pp149ndash165 1944

[26] J Wixted and D Rohrer ldquoAnalyzing the dynamics of free recallan integrative review of the empirical literaturerdquo PsychonomicBulletin and Review vol 1 pp 89ndash106 1994

[27] T D Frank and T Rhodes ldquoMicro-dynamic associated withtwo-state nonlinear Markov processes with an application tofree recallrdquo Fluctuation and Noise Letters vol 10 no 1 pp 41ndash58 2011

[28] T Rhodes andM T Turvey ldquoHumanmemory retrieval as Levyforagingrdquo Physica A vol 385 no 1 pp 255ndash260 2007

[29] J P Crutchfield and O Gornerup ldquoObjects that make objectsthe population dynamics of structural complexityrdquo Journal ofthe Royal Society Interface vol 3 no 7 pp 345ndash349 2006

[30] O Gonerup and J P Crutchfield ldquoHierachical self-organizationin the finitary process souprdquo Artificial Life vol 14 pp 245ndash2542008

[31] T D Frank ldquoStochastic processes and mean field systemsdefined by nonlinear Markov chains an illustration for a modelof evolutionary population dynamicsrdquo Brazilian Journal ofPhysics vol 41 no 2 pp 129ndash134 2011

[32] T D Frank ldquoNonlinear Markov processes deterministic caserdquoPhysics Letters A vol 372 no 41 pp 6235ndash6239 2008

[33] D Brogioli ldquoMarginally stable chemical systems as precursorsof liferdquo Physical Review Letters vol 105 no 5 Article ID 0581022010

[34] D Brogioli ldquoMarginally stability in chemical systems and itsrelevance in the origin of liferdquo Physical Review E vol 84 ArticleID 031931 2011

[35] T D Frank ldquoNonlinear physics approach to DNA cross-replication marginal stability generalized logistic growth andimpacts of degradationrdquo Physics Letters A vol 375 pp 3851ndash3857 2011

[36] Y Kuramoto Chemical Oscillations Waves and TurbulenceSpringer Berlin Germany 1984

[37] H Daido ldquoDiscrete-time population dynamics of interactingself-oscillatorsrdquo Progress ofTheoretical Physics vol 75 no 6 pp1460ndash1463 1986

[38] H Daido ldquoPopulation dynamics of randomly interacting self-oscillators I Tractable models without frustrationrdquo Progress ofTheoretical Physics vol 77 no 3 pp 622ndash634 1987

[39] H Daido ldquoScaling behaviour at the onset of mutual entrain-ment in a population of interacting oscillatorsrdquo Journal ofPhysics A vol 20 no 10 article 002 pp L629ndashL636 1987

[40] J N Teramae and Y Kuramoto ldquoStrong desynchronizing effectsof weak noise in globally coupled systemsrdquo Physical Review Evol 63 no 3 Article ID 36210 2001

[41] H Hara ldquoPath integrals for Fokker-Planck equation describedby generalized random walksrdquo Zeitschrift fur Physik B vol 45no 2 pp 159ndash166 1981

[42] E W Montroll and B J West ldquoModels of population growthdiffusion competition and rearrangementrdquo in SynergeticsCooperative Phenomena in Multi-Component Systems HakenEd pp 143ndash156 Springer Berlin Germany 1973

[43] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons New York NY USA 1977

[44] F D Nobre EM F Curado andG Rowlands ldquoA procedure forobtaining general nonlinear Fokker-Planck equationsrdquo PhysicaA vol 334 no 1-2 pp 109ndash118 2004

[45] T D Frank ldquoOn a nonlinear master equation and the Haken-Kelso-Bunz modelrdquo Journal of Biological Physics vol 30 no 2pp 139ndash159 2004

[46] D G Aronson ldquoThe porous medium equationrdquo in NonlinearDiffusion Problems Lecture Notes in Mathematics A Dobband B Eckmann Eds vol 1224 pp 1ndash46 Springer BerlinGermany 1986

[47] L A Peletier ldquoThe porous media equationrdquo in Applicationsof Nonlinear Analysis in the Physical Sciences H Amann NBazley and K Kirchgassner Eds vol 6 pp 229ndash241 PitmanAdvanced Publishing Program Boston Mass USA 1981

[48] N G van Kampen Stochastic Processes in Physics and Chem-istry vol 888 North-Holland Publishing Amsterdam TheNetherlands 1981

[49] S Kullback Information Theory and Statistics Dover Publica-tions Mineola NY USA 1968

[50] F Reif Fundamentals of Statistical and Thermal PhysicsMcGraw-Hill Book Company New York NY USA 1965

[51] E T Jaynes ldquoInformation theory and statistical mechanicsrdquoPhysical Review vol 106 no 4 pp 620ndash630 1957

[52] A Wehrl ldquoGeneral properties of entropyrdquo Reviews of ModernPhysics vol 50 no 2 pp 221ndash260 1978

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Page 24: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

24 ISRNMathematical Physics

[53] T D Frank ldquoNonextensive cutoff distributions of postural swayfor the old and the youngrdquo Physica A vol 388 no 12 pp 2503ndash2510 2009

[54] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[55] C Tsallis ldquoNon-extensive thermostatistics brief review andcommentsrdquo Physica A vol 221 no 1ndash3 pp 277ndash290 1995

[56] S Abe and Y Okamoto Nonextensive Statistical Mechanics andIts Applications Springer Berlin Germany 2001

[57] H Haken J A S Kelso and H Bunz ldquoA theoretical modelof phase transitions in human hand movementsrdquo BiologicalCybernetics vol 51 no 5 pp 347ndash356 1985

[58] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoTowards a comprehensive theory of brain activitycoupled oscillator systems under external forcesrdquoPhysicaD vol144 no 1-2 pp 62ndash86 2000

[59] G Schoner H Haken and J A S Kelso ldquoA stochastic theory ofphase transitions in human hand movementrdquo Biological Cyber-netics vol 53 no 4 pp 247ndash257 1986

[60] T D Frank ldquoLinear and nonlinear Fokker-Planck equationsrdquo inEncyclopedia of Complexity and Systems Science R A MeyersEd vol 5 pp 5239ndash5265 Springer Berlin Germany 2009

[61] J A Acebron L L Bonilla C J P Vicente F Ritort andR Spigler ldquoThe Kuramoto model a simple paradigm forsynchronization phenomenardquo Reviews of Modern Physics vol77 no 1 pp 137ndash185 2005

[62] H Haken Synergetics Introduction and Advanced TopicsSpringer Berlin Germany 2004

[63] P W Andersen ldquoMore is differentrdquo Science vol 177 no 4047pp 393ndash396 1972

[64] A Ichiki H Ito and M Shiino ldquoChaos-nonchaos phase tran-sitions induced by multiplicative noise in ensembles of coupledtwo-dimensional oscillatorsrdquo Physica E vol 40 no 2 pp 402ndash405 2007

[65] T Kanamaru ldquoBlowout bifurcation and on-off intermittencyin pulse neural networks with multiple modulesrdquo InternationalJournal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 11 pp 3309ndash3321 2006

[66] M Shiino and K Yoshida ldquoChaos-nonchaos phase transitionsinduced by external noise in ensembles of nonlinearly coupledoscillatorsrdquo Physical Review E vol 63 no 2 Article ID 026210pp 1ndash6 2001

[67] M A Zaks X Sailer L Schimansky-Geier and A B NeimanldquoNoise induced complexity from subthreshold oscillations tospiking in coupled excitable systemsrdquo Chaos vol 15 no 2Article ID 026117 2005

[68] R Balescu Equilibrium and Nonequilibrium Statistical Mechan-ics John Wiley amp Sons New York NY USA 1975

[69] R A Cairns Plasma Physics Blackie Philadelphia Pa USA1985

[70] J-L Delcroix Introduction to the Theory of Ionized GasesInterscience Publishers London UK 1960

[71] Y N Dnestrovskii andD P KostomarovNumerical Simulationsof Plasmas Springer Berlin Germany 1986

[72] E H Holt and R E Haskell Plasma Dynamics MacMillianNew York NY USA 1965

[73] Yu L Klimontovich Statistical Physics Harwood AcademicPublishers New York NY USA 1986

[74] D R Nicholson Introduction to PlasmaTheory JohnWiley andSons New York NY USA 1983

[75] M Soler F C Martınez and J M Donoso ldquoIntegral kineticmethod for one dimension the spherical caserdquo Journal ofStatistical Physics vol 69 no 3-4 pp 813ndash835 1992

[76] E J Allen and H D Victory ldquoA computational investigationof the random particle method for numerical solution of thekinetic Vlasov-Poisson-Fokker-Planck equationsrdquo Physica Avol 209 no 3-4 pp 318ndash346 1994

[77] W M MacDonald M N Rosenbluth and W Chuck ldquoRelax-ation of a system of particles with coulomb interactionsrdquoPhysical Review vol 107 no 2 pp 350ndash353 1957

[78] M N Rosenbluth W M MacDonald and D L Judd ldquoFokker-Planck equation for an inverse-square forcerdquo vol 107 pp 1ndash61957

[79] M Takai H Akiyama and S Takeda ldquoStabilization of drift-cyclotron loss-cone instability of plasmas by high frequencyfieldrdquo Journal of the Physical Society of Japan vol 50 no 5 pp1716ndash1722 1981

[80] B M Boghosian ldquoThermodynamic description of the relax-ation of two-dimensional turbulence using Tsallis statisticsrdquoPhysical Review E vol 53 no 5 pp 4754ndash4763 1996

[81] PHChavanis ldquoNonlinearmeanfield Fokker-Planck equationsApplication to the chemotaxis of biological populationsrdquo Euro-pean Physical Journal B vol 62 no 2 pp 179ndash208 2008

[82] P H Chavanis and C Sire ldquoAnomalous diffusion and collapseof self-gravitating Langevin particles in D dimensionsrdquo PhysicalReview E vol 69 no 1 Article ID 016116 2004

[83] A R Plastino and A Plastino ldquoStellar polytropes and Tsallisrsquoentropyrdquo Physics Letters A vol 174 no 5-6 pp 384ndash386 1993

[84] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[85] J Binney and S TremaineGalatic Dynamics PrincetonUniver-sity Press Princeton NJ USA 1987

[86] C Lancellotti and M Kiessling ldquoSelf-similar gravitationalcollapse in stellar dynamicsrdquo Astrophysical Journal vol 549 ppL93ndashL96 2001

[87] A Chao Physics of Collective Instabilities in High Energy Accel-erators John Wiley and Sons New York NY USA 1993

[88] H Wiedemann Particle Accelerator Physics II Nonlinear andHigher Order Dynamics Springer Berlin Germany 1993

[89] R C Davidson H Qin and T S F Wang ldquoVlasov-Maxwelldescription of electron-ion two-stream instability in high-intensity linacs and storage ringsrdquo Physics Letters A vol 252no 5 pp 213ndash221 1999

[90] R C Davidson H Qin and P J Channell ldquoPeriodically-focused solutions to the nonlinear Vlasov-Maxwell equationsfor intense charged particle beamsrdquo Physics Letters A vol 258no 4ndash6 pp 297ndash304 1999

[91] S Heifets ldquoMicrowave instability beyond thresholdrdquo PhysicalReview Special TopicsmdashAccelerators and Beams vol 54 pp2889ndash2898 1996

[92] S Heifets ldquoSaturation of the coherent beam-beam instabilityrdquoPhysical Review Special TopicsmdashAccelerators and Beams vol 4no 4 Article ID 044401 pp 98ndash105 2001

[93] S Heifets ldquoSingle-mode coherent synchrotron radiation insta-bility of a bunched beamrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 6 no 8 Article ID 080701 pp 10ndash21 2003

[94] S Heifets and B Podobedov ldquoSingle bunch stability to monop-ole excitationrdquo Physical Review Special TopicsmdashAccelerators andBeams vol 2 no 4 Article ID 044402 pp 43ndash49 1999

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Page 25: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 25

[95] Y Shobuda and K Hirata ldquoThe existence of a static solution ofthe Haissinski equation with purely inductive wake forcerdquo PartAccel vol 62 pp 165ndash177 1999

[96] Y Shobuda and K Hirata ldquoProof of the existence and unique-ness of a solution for the Haissinski equation with a capacitivewake functionrdquo Physical Review E vol 64 no 6 Article ID067501 2001

[97] G V Stupakov B N Breizman and N S Pekker ldquoNonlineardynamics of microwave instability in acceleratorsrdquo PhysicalReview E vol 55 pp 5976ndash5984 1997

[98] M Venturini and R Warnock ldquoBursts of coherent synchrotronradiation in electron storage rings a dynamicalmodelrdquo PhysicalReview Letters vol 89 no 22 Article ID 224802 2002

[99] J Haıssinski ldquoExact longitudinal equilibrium distribution ofstored electrons in the presence of self-fieldsrdquo Nuovo CimentoDella Societa Italiana Di Fisica B vol 18 no 1 pp 72ndash82 1973

[100] T D Frank ldquoSmoluchowski approach to nonlinear Vlasov-Fokker-Planck equations Stability analysis of beam dynam-ics and Haıssinski theoryrdquo Physical Review Special TopicsmdashAccelerators and Beams vol 9 no 8 Article ID 084401 2006

[101] T D Frank and S Mongkolsakulvong ldquoA nonextensive ther-mostatistical approach to the Haıssinski theory of acceleratorbeamsrdquo Physica A vol 387 no 19-20 pp 4828ndash4838 2008

[102] T D Frank and S Mongkolsakulvong ldquoParametric solutionmethod for self-consistency equations and order parameterequations derived from nonlinear Fokker-Planck equationsrdquoPhysica D vol 238 no 14 pp 1186ndash1196 2009

[103] C Thomas R Bartolini J I M Botman G Dattoli L Meziand M Migliorati ldquoAn analytical solution for the Haissinskiequation with purely inductive wake fieldsrdquo Europhysics Lettersvol 60 no 1 pp 66ndash71 2002

[104] M Venturini ldquoStability analysis of longitudinal beam dynamicsusing noncanonical Hamiltonian methods and energy princi-plesrdquo Physical Review Special TopicsmdashAccelerators and Beamsvol 5 no 5 pp 43ndash50 2002

[105] S Chandrasekhar Liquid Crystals Cambridge University PressCambridge UK 1977

[106] P De Gennes The Physics of Liquid Crystals Clarendon PressOxford UK 1974

[107] M Plischke and B Bergersen Equilibrium Statistical PhysicsWorld Scientifric Singapore 1994

[108] G Strobl Condensed Matter Physics Crystals Liquids LiquidCrystals and Polymers Springer Berlin Germany 2004

[109] W Maier and A Saupe ldquoEine einfache molekulare Theoriedes namatischen kristallinflussigen Zustandesrdquo Zeitschrift FurNaturforschung A vol 13 pp 564ndash566 1958 (German)

[110] W Maier and A Saupe ldquoEine einfache molekulare Theorie desnamatischen kristallinflussigen Zustandes Teil IIrdquo ZeitschriftFur Naturforschung A vol 15 pp 287ndash292 1960 (German)

[111] A Saupe ldquoLiquid crystalsrdquo Annual Review of Physical Chem-istry vol 24 pp 441ndash471 1973

[112] C V Hess ldquoFokker-Planck equation approach to flow align-ment in liquid crystalsrdquo Zeitschrift Fur Naturforschung vol 31pp 1034ndash1037 1976

[113] M Doi and S F Edwards The Theory of Polymer DynamicsClarendon Press Oxford UK 1988

[114] T D Frank ldquoMaier-Saupe model of liquid crystals Isotropic-nematic phase transitions and second-order statistics studiedby Shiinorsquos perturbation theory and strongly nonlinear Smolu-chowski equationsrdquo Physical Review E vol 72 no 4 Article ID041703 2005

[115] P Ilg I V Karlin and H C Ottinger ldquoGenerating momentequations in the Doi model of liquid-crystalline polymersrdquoPhysical Review E vol 60 pp 5783ndash5787 1999

[116] F Verhulst Nonlinear Differential Equations and DynamicalSystems Springer Berlin Germany 1996

[117] B U Felderhof ldquoOrientational relaxation in the Maier-Saupemodel of nematic liquid crystalsrdquo Physica A vol 323 no 1ndash4pp 88ndash106 2003

[118] B U Felderhof and R B Jones ldquoMean field theory of the non-linear response of an interacting dipolar system with rotationaldiffusion to an oscillating fieldrdquo Journal of Physics CondensedMatter vol 15 no 23 pp 4011ndash4024 2003

[119] P Ilg M Kroger S Hess and A Y Zubarev ldquoDynamics ofcolloidal suspensions of ferromagnetic particles in plane Cou-ette flow comparison of approximate solutions with Browniandynamics simulationsrdquo Physical Review E vol 67 Article ID061401 2003

[120] R G Larson and H C Ottinger ldquoEffect of molecular elas-ticity on out-of-plane orientations in shearing flows of liquid-crystalline polymersrdquoMacromolecules vol 24 no 23 pp 6270ndash6282 1991

[121] HC Ottinger Stochastic Processes in Polymeric Fluids SpringerBerlin Germany 1996

[122] G Kaniadakis ldquo119867-theorem and generalized entropies withinthe framework of nonlinear kineticsrdquo Physics Letters A vol 288no 5-6 pp 283ndash291 2001

[123] G Kaniadakis and P Quarati ldquoKinetic equation for classicalparticles obeying an exclusion principlerdquo Physical Review E vol48 no 6 pp 4263ndash4270 1993

[124] G Kaniadakis and P Quarati ldquoClassical model of bosons andfermionsrdquo Physical Review E vol 49 no 6 pp 5103ndash5110 1994

[125] E A Uehling and G E Uhlenbeck ldquoTransport phenomena inEinstein-Bose or Fermi-Dirac gas IrdquoPhysical Review vol 43 pp552ndash561 1933

[126] T D Frank ldquoClassical Langevin equations for the free electrongas and blackbody radiationrdquo Journal of Physics A vol 37 no 11pp 3561ndash3567 2004

[127] T D Frank ldquoModelling the stochastic single particle dynamicsof relativistic fermions and bosons using nonlinear drift-diffusion equationsrdquo Mathematical and Computer Modellingvol 42 no 9-10 pp 1057ndash1062 2005

[128] T D Frank and A Daffertshofer ldquoNonlinear Fokker-Planckequations whose stationary solutions make entropy-like func-tionals stationaryrdquo Physica A vol 272 no 3-4 pp 497ndash5081999

[129] T D Frank and A Daffertshofer ldquoMultivariate nonlinearFokker-Planck equations and generalized thermostatisticsrdquoPhysica A vol 292 no 1ndash4 pp 392ndash410 2001

[130] L P Kadanoff Statistical Physics Statics Dynamics and Renor-malization World Scientific Publishing River Edge NJ USA2000

[131] J Sopik C Sire and P-H Chavanis ldquoDynamics of the Bose-Einstein condensation analogy with the collapse dynamics of aclassical self-gravitating Brownian gasrdquo Physical Review E vol74 no 1 Article ID 011112 p 15 2006

[132] R Tsekov ldquoDissipation in quantum systemsrdquo Journal of PhysicsA vol 28 no 21 pp L557ndashL561 1995

[133] R Tsekov ldquoA quantum theory of thermodynamic relaxationrdquoInternational Journal of Molecular Sciences vol 2 no 2 pp 66ndash71 2001

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 26: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

26 ISRNMathematical Physics

[134] W Ebeling ldquoCanonical nonequilibrium statistics and applica-tions to Fermi-Bose systemsrdquo Condensed Matter Physics vol 3pp 285ndash293 2000

[135] WM Ni L A Peletier and J SerrinNonlinear Diffusion Equa-tions and Their Equilibrium States Springer Berlin Germany1988

[136] G I Barenblatt V M Entov and V M RyzhikTheory of FluidFlows through Natural Rocks Kluwer Academic PublishersDordrecht The Netherlands 1990

[137] R E Pattle ldquoDiffusion from an instantaneous point source witha concentration-dependent coefficientrdquo The Quarterly Journalof Mechanics and Applied Mathematics vol 12 pp 407ndash4091959

[138] L Borland ldquoMicroscopic dynamics of the nonlinear Fokker-Planck equationrdquo Physical Review E vol 57 no 6 pp 6634ndash6642 1996

[139] A Compte and D Jou ldquoNon-equilibrium thermodynamics andanomalous diffusionrdquo Journal of Physics A vol 29 no 15 pp4321ndash4329 1996

[140] C Tsallis and D J Bukman ldquoAnomalous diffusion in thepresence of external forces Exact time-dependent solutions andtheir thermostatistical basisrdquo Physical Review E vol 54 no 3pp R2197ndashR2200 1996

[141] T D Frank ldquoAutocorrelation functions of nonlinear Fokker-Planck equationsrdquo European Physical Journal B vol 37 no 2pp 139ndash142 2004

[142] J S Andrade G F T Da Silva A A Moreira F D Nobre andE M F Curado ldquoThermostatistics of overdamped motion ofinteracting particlesrdquo Physical Review Letters vol 105 no 26Article ID 260601 2010

[143] M S Ribeiro F D Nobre and EM F Curado ldquoTime evolutionof interacting vortices under overdamped motionrdquo PhysicalReview E vol 85 Article ID 021146 2012

[144] M S Ribeiro F D Nobre and E M F Curado ldquoOverdampedmotion of interacting particles in general confining poten-tials time-dependent and stationary-state analysisrdquo EuropeanPhysics Journal B vol 85 article 399 2012

[145] A R Plastino and A Plastino ldquoNon-extensive statisticalmechanics and generalized Fokker-Planck equationrdquo Physica Avol 222 no 1ndash4 pp 347ndash354 1995

[146] S Martinez A R Plastino and A Plastino ldquoNonlinear Fokker-Planck equations and generalized entropiesrdquo Physica A vol 259no 1-2 pp 183ndash192 1998

[147] T D Frank ldquoA Langevin approach for the microscopic dynam-ics of nonlinear Fokker-Planck equationsrdquo Physica A vol 301no 1ndash4 pp 52ndash62 2001

[148] G Drazer H S Wio and C Tsallis ldquoAnomalous diffusion withabsorption exact time-dependent solutionsrdquo Physical Review Evol 61 no 2 pp 1417ndash1422 2000

[149] A R Plastino M Casas and A Plastino ldquoA nonextensivemaximum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 pp 289ndash303 2000

[150] A Rigo A R Plastino M Casas and A Plastino ldquoAnomalousdiffusion coupled with Verhulst-like growth dynamics exacttime-dependent solutionsrdquo Physics Letters A vol 276 no 1ndash4pp 97ndash102 2000

[151] E K Lenzi C Anteneodo and L Borland ldquoEscape time inanomalous diffusive mediardquo Physical Review E vol 63 no 5 IArticle ID 51109 2001

[152] A Compte D Jou and Y Katayama ldquoAnomalous diffusion inlinear shear flowsrdquo Journal of Physics A vol 30 no 4 pp 1023ndash1030 1997

[153] L C Malacarne R S Mendes and I T Pedron ldquoNonlin-ear equation for anomalous diffusion unified power-law andstretched exponential exact solutionrdquo Physical Review E vol 63no 3 I Article ID 030101 2001

[154] I T Pedron R S Mendes L C Malacarne and E K LenzildquoNonlinear anomalous diffusion equation and fractal dimen-sion exact generalized Gaussian solutionrdquo Physical Review Evol 65 no 4 Article ID 041108 2002

[155] L Borland F Pennini A R Plastino andA Plastino ldquoThenon-linear Fokker-Planck equation with state-dependent diffusiona nonextensivemaximumentropy approachrdquoEuropean PhysicalJournal B vol 12 no 2 pp 285ndash297 1999

[156] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 129ndash144 2000

[157] P H Chavanis ldquoGeneralized thermodynamics and Fokker-Planck equations applications to stellar dynamics and two-dimensional turbulencerdquo Physical Review E vol 68 no 3Article ID 036108 2003

[158] P-H Chavanis ldquoGeneralized Fokker-Planck equations andeffective thermodynamicsrdquo Physica A vol 340 no 1ndash3 pp 57ndash65 2004

[159] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[160] M S Ribeiro F D Nobre and E M F Curado ldquoClasses of119873-dimensional nonlinear Fokker-Planck equations associated toTsallis entropyrdquo Entropy vol 13 no 11 pp 1928ndash1944 2011

[161] V Schwammle F D Nobre and E M F Curado ldquoConse-quences of the H-theorem for nonlinear Fokker-Planck equa-tionsrdquo Physical Review E vol 76 Article ID 041123 2007

[162] V Schwammle E M F Curado and F D Nobre ldquoA generalnonlinear Fokker-Planck equation and its associated entropyrdquoTheEuropean Physical Journal B vol 58 no 2 pp 159ndash165 2007

[163] V Schwammle E M F Curado and F D Nobre ldquoDynamicsof normal and anomalous diffusion in nonlinear Fokker-Planckequationsrdquo The European Physical Journal B vol 70 no 1 pp107ndash116 2009

[164] M Shiino ldquoFree energies based on generalized entropies and119867-theorems for nonlinear Fokker-Planck equationsrdquo Journal ofMathematical Physics vol 42 no 6 pp 2540ndash2553 2001

[165] S Mongkolsakulvong and T D Frank ldquoCanonical-dissipativelimit cycle oscillators with a short-range interaction in phasespacerdquoCondensedMatter Physics vol 13 no 1 Article ID 13001pp 1ndash18 2010

[166] W Ebeling and I M Sokolov Statistical thermodynamics andstochastic theory of nonequilibrium systems World ScientificPublishing Singapore 2004

[167] H Haken ldquoDistribution function for classical and quantumsystems far from thermal equilibriumrdquo Zeitschrift fur Physikvol 263 no 4 pp 267ndash282 1973

[168] F Schweitzer Brownian Agents and Active Particles SpringerBerlin Germany 2003

[169] A Pikovsky M Rosenblum and J Kurths SynchronizationA Universal Concept in Nonlinear Sciences vol 12 CambridgeUniversity Press Cambridge UK 2001

[170] T D Frank and M J Richardson ldquoOn a test statistic for theKuramoto order parameter of synchronization an illustrationfor group synchronization during rocking chairsrdquo Physica Dvol 239 no 23-24 pp 2084ndash2092 2010

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 27: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

ISRNMathematical Physics 27

[171] M J Richardson R L Garcia T D Frank M Gregor and K LMarsh ldquoMeasuring group synchrony a cluster-phase methodfor analyzing multivariate movement time seriesrdquo Frontiers inPhysiology vol 3 article 405 2012

[172] S H Strogatz ldquoFrom Kuramoto to Crawford exploring theonset of synchronization in populations of coupled oscillatorsrdquoPhysica D vol 143 no 1ndash4 pp 1ndash20 2000

[173] T D Frank A Daffertshofer C E Peper P J Beek and HHaken ldquoH-theorem for a mean field model describing coupledoscillator systems under external forcesrdquo Physica D vol 150 no3-4 pp 219ndash236 2001

[174] T D Frank ldquoOn a mean field Haken-Kelso-Bunz model anda free energy approach to relaxation processesrdquo NonlinearPhenomena in Complex Systems vol 5 no 4 pp 332ndash341 2002

[175] P A Tass Phase Resetting in Medicine and Biology StochasticModelling and Data Analysis Springer Berlin Germany 1999

[176] A Okubo and S A Levin Diffusion and Ecological ProblemsModern Perspectives Springer New York NY USA 2001

[177] A Okubo Diffusion and Ecological Problems MathematicalModels Springer Berlin Germany 1980

[178] J D MurrayMathematical Biology Springer Berlin Germany1993

[179] T P Witelski ldquoSegregation and mixing in degenerate diffusionin population dynamicsrdquo Journal of Mathematical Biology vol35 no 6 pp 695ndash712 1997

[180] F J Boster J E Hunter M E Mayer and J L Hale ldquoExpandingthe persuasive arguments explanation of the polarity shift alinear discrepancy modelrdquo in Communication Yearbook 4 DNimmo Ed pp 165ndash176 Transaction Books New BrunswickCanada 1980

[181] F J Boster J E Fryrear P A Mongeau and J E Hunter ldquoAnunequal speaking linear discrepancy model implications forpolarity shiftrdquo in Communication Yearbook 6 M Burgoon Edpp 395ndash418 Sage Publishers Beverly Hills Calif USA 1982

[182] F J Boster J E Hunter and J L Hale ldquoAn informationprocessing model of jury decision makingrdquo CommunicationResearch vol 18 pp 524ndash547 1991

[183] T D Frank ldquoOn the linear discrepancy model and risky shiftsin group behavior a nonlinear Fokker-Planck perspectiverdquoJournal of Physics A vol 42 no 15 Article ID 155001 2009

[184] L Borland ldquoOption pricing formulas based on a non-Gaussianstock price modelrdquo Physical Review Letters vol 89 no 9 pp987011ndash987014 2002

[185] L Borland ldquoNon-Gaussian option pricing Successes limita-tions and perspectivesrdquo in Anomalous Fluctuations in ComplexSystems Plasmas Fluids and Financial Markets C Riccardi andH E Roman Eds pp 311ndash334 2008

[186] L Borland and J-P Bouchaud ldquoA non-Gaussian option pricingmodel with skewrdquo Quantitative Finance vol 4 no 5 pp 499ndash514 2004

[187] C F Lo and C H Hui ldquoA simple analytical model for dynamicsof time-varying target leverage ratiosrdquo European Physics JournalB vol 85 article 102 2012

[188] C H Hui C F Lo and M X Huang ldquoAre corporatesrsquotarget leverage ratios time-dependentrdquo International Review ofFinancial Analysis vol 15 no 3 pp 220ndash236 2006

[189] S Primak ldquoGeneration of compound non-Gaussian randomprocesses with a given correlation functionrdquo Physical Review Evol 61 no 1 pp 100ndash103 2000

[190] S Primak V Lyandres and V Kontorovich ldquoMarkov modelsof non-Gaussian exponentially correlated processes and theirapplicationsrdquo Physical Review E vol 63 no 6 I Article ID061103 2001

[191] H Shimizu ldquoMuscle contraction mechanism as a hard modeinstabilityrdquo Progress of Theoretical Physics vol 52 pp 329ndash3301974

[192] H Shimizu and T Yamada ldquoPhenomenological equations ofmotion of muscular contrationsrdquo Progress ofTheoretical Physicsvol 47 pp 350ndash351 1972

[193] T D Frank ldquoA note on the Markov property of stochastic proc-esses described by nonlinear Fokker-Planck equationsrdquo PhysicaA vol 320 no 1ndash4 pp 204ndash210 2003

[194] R C Desai and R Zwanzig ldquoStatistical mechanics of a nonlin-ear stochastic modelrdquo Journal of Statistical Physics vol 19 no 1pp 1ndash24 1978

[195] D A Dawson ldquoCritical dynamics and fluctuations for a mean-field model of cooperative behaviorrdquo Journal of StatisticalPhysics vol 31 no 1 pp 29ndash85 1983

[196] M Shiino ldquoDynamical behavior of stochastic systems of infi-nitelymany coupled nonlinear oscillators exhibiting phase tran-sitions of mean-field type H-theorem on asymptotic approachto equilibrium and critical slowing down of order-parameterfluctuationsrdquo Physical Review A vol 36 no 5 pp 2393ndash24121987

[197] A Arnold L L Bonilla and P A Markowich ldquoLiapunovfunctionals and large-time-asymptotics of mean-field nonlin-ear Fokker-Planck equationsrdquo Transport Theory and StatisticalPhysics vol 25 no 7 pp 733ndash751 1996

[198] L L Bonilla C J Perez-Vicente F Ritort and J Soler ldquoExactlysolvable phase oscillator models with synchronization dynam-icsrdquo Physical Review Letters vol 81 no 17 pp 3643ndash3646 1998

[199] G Kaniadakis ldquoNonlinear kinetics underlying generalizedstatisticsrdquo Physics Letters A vol 296 pp 405ndash425 2001

[200] J A S Lima R Silva and A R Plastino ldquoNonextensivethermostatistics and the H theoremrdquo Physical Review Lettersvol 86 no 14 pp 2938ndash2941 2001

[201] M Shiino ldquoNonlinear Fokker-Planck equation exhibiting bifur-cation phenomena and generalized thermostatisticsrdquo Journal ofMathematical Physics vol 43 no 5 pp 2654ndash2669 2002

[202] T D Frank ldquoDynamic mean field models H-theorem forstochastic processes and basins of attraction of stationaryprocessesrdquo Physica D vol 195 no 3-4 pp 229ndash243 2004

[203] L L Bonilla ldquoStable nonequilibrium probability densities andphase transitions for meanfield models in the thermodynamiclimitrdquo Journal of Statistical Physics vol 46 no 3-4 pp 659ndash6781987

[204] J D Crawford and K T R Davies ldquoSynchronization of globallycoupled phase oscillators singularities and scaling for generalcouplingsrdquo Physica D vol 125 no 1-2 pp 1ndash46 1999

[205] V N Kolokoltsov Nonlinear Markov Processes and KineticEquations Cambridge University Press Cambridge UK 2010

[206] W Braun and K Hepp ldquoThe Vlasov dynamics and its fluc-tuations in the 1119873 limit of interacting classical particlesrdquoCommunications inMathematical Physics vol 56 no 2 pp 101ndash113 1977

[207] T Funaki ldquoA certain class of diffusion processes associatedwith nonlinear parabolic equationsrdquo Zeitschrift fur Wahrschein-lichkeitstheorie und Verwandte Gebiete vol 67 no 3 pp 331ndash348 1984

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 28: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

28 ISRNMathematical Physics

[208] J Gartner ldquoOn the McKean-Vlasov limit for interacting diffu-sionsrdquoMathematische Nachrichten vol 137 pp 197ndash248 1988

[209] S Meleard and S Roelly-Coppoletta ldquoA propagation of chaosresult for a system of particles with moderate interactionrdquoStochastic Processes and their Applications vol 26 no 2 pp 317ndash332 1987

[210] K Oelschlager ldquoA law of large numbers formoderately interact-ing diffusion processesrdquo Zeitschrift fur Wahrscheinlichkeitstheo-rie und Verwandte Gebiete vol 69 no 2 pp 279ndash322 1985

[211] A Arnold P Markowich and G Toscani ldquoOn large timeasymptotics for drift-diffusion-Poisson systemsrdquoTransportThe-ory and Statistical Physics vol 29 no 3ndash5 pp 571ndash581

[212] A Arnold P Markowich G Toscani and A Unterreiter ldquoOnconvex Sobolev inequalities and the rate of convergence to equi-librium for Fokker-Planck type equationsrdquo Communications inPartial Differential Equations vol 26 no 1-2 pp 43ndash100 2001

[213] J A Carrillo A Jungel P A Markowich G Toscani andA Unterreiter ldquoEntropy dissipation methods for degener-ate parabolic problems and generalized Sobolev inequalitiesrdquoMonatshefte fur Mathematik vol 133 no 1 pp 1ndash82 2001

[214] D A Dawson and J Gartner ldquoLarge deviations free energyfunctional and quasi-potential for a mean field model ofinteracting diffusionsrdquo Memoirs of the American MathematicalSociety vol 78 no 398 1989

[215] B Djehiche and I Kaj ldquoThe rate function for some measure-valued jump processesrdquoThe Annals of Probability vol 23 no 3pp 1414ndash1438 1995

[216] J Fontbona ldquoNonlinearmartingale problems involving singularintegralsrdquo Journal of Functional Analysis vol 200 no 1 pp 198ndash236 2003

[217] C Graham ldquoNonlinear limit for a system of diffusing particleswhich alternate between two statesrdquo Applied Mathematics andOptimization vol 22 no 1 pp 75ndash90 1990

[218] A Greven ldquoRenormalization and universality for multitypepopulation modelsrdquo in Interacting Stochastic Systems pp 209ndash246 Springer Berlin Germany 2005

[219] L Overbeck ldquoNonlinear superprocessesrdquoThe Annals of Proba-bility vol 24 no 2 pp 743ndash760 1996

[220] B Jourdain ldquoDiffusion Processes Associated with NonlinearEvolution Equations for Signed Measuresrdquo Methodology andComputing in Applied Probability vol 2 pp 69ndash91 2000

[221] D A Dawson ldquoMeasure-Valued Markov Processesrdquo in LectureNotes in Mathematics D A Dawson B Maisonneuve and JSpencer Eds vol 1541 Springer Berlin Germany 1993

[222] D W Stroock Markov Processes from K Itorsquos Perspective vol155 Princeton University Press Princeton NJ USA 2003

[223] A Friedman Partial Differential Equations of Parabolic TypePrentice-Hall Englewood Cliffs NJ USA 1964

[224] A Friedman Partial Differential Equations Holt Richart andWinston New York NY USA 1969

[225] J D Logan Transport Modeling in Hydrogeochemical SystemsSpringer New York NY USA 2001

[226] G NMilstein ldquoThe probability approach to numerical solutionof nonlinear parabolic equationsrdquo Numerical Methods for Par-tial Differential Equations vol 18 no 4 pp 490ndash522 2002

[227] G N Milstein and M V Tretyakov ldquoNumerical methods fornonlinear parabolic equations with small parameter based onprobability approachrdquoMathematics of Computation vol 60 pp237ndash267 2000

[228] G N Milstein and M V Tretyakov ldquoNumerical solution ofthe Dirichlet problem for nonlinear parabolic equations by aprobabilistic approachrdquo IMA Journal of Numerical Analysis vol21 no 4 pp 887ndash917 2001

[229] DHuber and L S Tsimring ldquoDynamics of an ensemble of noisybistable elements with global time delayed couplingrdquo PhysicalReview Letters vol 91 no 26 I Article ID 260601 2003

[230] S Kim S H Park and H B Pyo ldquoStochastic resonance incoupled oscillator systems with time delayrdquo Physical ReviewLetters vol 82 pp 1620ndash1623 1999

[231] E Niebur H G Schuster and D M Kammen ldquoCollective fre-quencies andmetastability in networks of limit-cycle oscillatorswith time delayrdquo Physical Review Letters vol 67 no 20 pp2753ndash2756 1991

[232] M K S Yeung and S H Strogatz ldquoTime delay in the kuramotomodel of coupled oscillatorsrdquo Physical Review Letters vol 82no 3 pp 648ndash651 1999

[233] T D Frank and P J Beek ldquoFokker-Planck equations forglobally coupled many-body systems with time delaysrdquo Journalof Statistical Mechanics vol 10 Article ID P10010 2005

[234] N S Goel and N Richter-Dyn Stochastic Models in BiologyAcademic Press New York NY USA 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 29: Review Article Strongly Nonlinear Stochastic Processes in ...downloads.hindawi.com/archive/2013/149169.pdf · Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of