Exact Partition Function for the Random Walk of an ...downloads.hindawi.com/journals/amp/2017/6970870.pdfExact Partition Function for the Random Walk of an Electrostatic Field GabrielGonzález1,2

Research ArticleExact Partition Function for the Random Walk ofan Electrostatic Field

Gabriel Gonzaacutelez12

1Catedras CONACYT Universidad Autonoma de San Luis Potosı 78000 San Luis Potosı SLP Mexico2Coordinacion para la Innovacion y la Aplicacion de la Ciencia y la Tecnologıa Universidad Autonoma de San Luis Potosı78000 San Luis Potosı SLP Mexico

Correspondence should be addressed to Gabriel Gonzalez gabrielgonzalezuaslpmx

Received 7 April 2017 Revised 6 June 2017 Accepted 14 June 2017 Published 13 July 2017

Academic Editor Jacopo Bellazzini

Copyright copy 2017 Gabriel Gonzalez This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The partition function for the random walk of an electrostatic field produced by several static parallel infinite charged planes inwhich the charge distribution could be either plusmn120590 is obtained We find the electrostatic energy of the system and show that it canbe analyzed through generalized Dyck paths The relation between the electrostatic field and generalized Dyck paths allows us tosum overall possible electrostatic field configurations and is used for obtaining the partition function of the system We illustrateour results with one example

1 Introduction

In 1961 Lenard considered the problem of a system of infinitecharged planes free to move in one direction without anyinhibition of free crossing over each other [1] If all thecharged planes carry a surface mass density of magnitudeunity and the 119894th one carries an electric surface charge density120590119894 the Hamiltonian of Lenardrsquos problem is given by

H (1199021 1199022 1199011 1199012 )= 12sum119894 119901

2119894 minus 121205980 sumsum

119894lt119895

120590119894120590119895 10038161003816100381610038161003816119902119894 minus 11990211989510038161003816100381610038161003816 (1)

where the system is overall neutral With the Hamiltonianfunction at handLenardwas able to obtain the exact statisticalmechanics of the system with the technique of generatingfunctions [1 2]

From a physical point of view Lenardrsquosmodel can be usedin the study of a one-dimensional Coulomb gas [3] The one-dimensional Coulomb gas is a statistical mechanical problemwhere particles of equal or opposite charges interact throughthe Coulomb potential [4] The model has been extensivelystudied in the past and forms one of the classical exactlysoluble problems in one dimension [5] Also the Coulomb

gas model has also been used to describe the main features ofionic liquids [6 7]

More recently a very similar problemwas proposed in [8]with the constraint that each of the charged planes has a fixedposition in space and that the surface charge distribution ineach plane could be either plusmn120590This modifiedmodel gives riseto a random walk behavior of the electrostatic field that canbe analyzed as a Markovian stochastic process

Thepurpose of this paper is to give a statistical descriptionof the electrostatic field generated by several static parallelinfinite charged planes in which the surface charge distri-bution could be either plusmn120590 The key step in our formulationconsists of the summation of all possible trajectories of theelectrostatic field for every different charge configurationWedo this by showing that there is a one to one correspondencebetween every electrostatic field trajectory and a generalizedDyck path

The article is organized as follows In Section 2 wedescribe the model and derive a system of equations forobtaining the electrostatic energy of the system or Hamil-tonian In Section 3 we make a one to one correspondencebetween electrostatic field trajectories and generalized Dyckpaths In Section 4 the explicit expression for the partitionfunction is given In Section 5 a simple numerical analysis is

HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 6970870 5 pageshttpsdoiorg10115520176970870

2 Advances in Mathematical Physics

z

Figure 1 Possible electrostatic field configurations inside a four-charged-plane system where the charge distribution is not known

given to show some features of the partition function In thelast section we summarize our conclusions

2 Model of the System

Suppose that we are given a collection of 2119873 infinite chargedplanes half of them have a constant charge distribution 120590and the other half have a constant charge distribution minus120590that is the system is overall neutral If we randomly place the2119873 infinite charged planes parallel to each other along the 119911-axis at position 119911119899 = 119899119889 where 119899 isin 0 1 2119873 minus 1 thenthe electrostatic field would evolve along the 119911-axis makingrandom jumps each time it crosses an infinite charged sheet

For example consider a system of four charged planesWe know that 119864(119911) equals zero at the left and right side ofthe configuration due to the neutrality of the system [9]After crossing the first charged plane 119864(119911) would increaseor decrease in an amount of 1205901205980 depending on whether thefirst charged plane had a positive or negative surface chargedensity In Figure 1 we show all the possible electrostatic fieldconfigurations for the case when the charge density is notknown

Since we are concern with the statistical properties of theelectrostatic field we have first to obtain the Hamiltonian ofthe field or the electrostatic energy of the system We knowfrom basic electrodynamics courses that the electrostaticenergy is given by [9]

E = 120575119864Δ119881 (2)

where 120575119864 = 120598011986422 is the electrostatic energy density andΔ119881 = 119860119889 is the volume inside the region between twoconsecutive parallel charged sheets Denoting 119864(119894) as thevalue of the electrostatic field in the 119894th region between twocharged sheets placed in 119911 = (119894minus1)119889 and 119911 = 119894119889 then we havethe following boundary condition

119864 (119894) minus 119864 (119894 minus 1) = 1205901205980 119878119894 for 119894 = 1 2 2119873 (3)

where 119878119894 is a random variable which takes one of the twopossible values plusmn1 depending on whether the charged planeat 119911 = (119894 minus 1)119889 is positively or negatively charged

We can write down the boundary condition given in (3)in matrix form in the following way

[[[[[[[[[[

1 0 0 0 0minus1 1 0 0 00 minus1 1 0 0 d d

0 sdot sdot sdot minus1 1

]]]]]]]]]]

[[[[[[[[[[

119864 (0)119864 (1)119864 (2)

119864 (2119873)

]]]]]]]]]]

= 1205901205980[[[[[[[[[[

0119878111987821198782119873

]]]]]]]]]]

(4)

where we have included the initial condition of the electro-static field that is 119864(0) = 0 Solving (4) for the electrostaticfield we have

119864 (119894) = 1205901205980119894sum119895=1

119878119895 for 119894 = 1 2 2119873 (5)

Substituting (5) into (2) we obtain the electrostatic energy ofthe system which is given by

E = (12059802 119860119889sum119894 1198642 (119894)) 120575sum2119873

119894=1119878119894 0 (6)

where the Kronecker delta guarantees the neutrality of thesystem

3 Electrostatic Field and Dyck Paths

In this section we present the relation between the electro-static field and generalizedDyck paths Let us consider for thesake of simplicity that all the charged planes carry a surfacecharge density of 120590 = plusmn1119860 Let 119876119894 be the total amount ofcharge to the left of the 119894th region then we can rewrite theelectrostatic field given in (5) as

119864 (119894) = 1198761198941205980119860 (7)

Note that1198760 = 1198762119873 = 0 and |119876119894 minus119876119894minus1| = 1 The electrostaticenergy in terms of the new variable is given then by

E = 12119862sum119894 1198762119894 (8)

where 119862 = 1205980119860119889It is convenient to investigate the nature of all the possible

electrostatic field configurations by plotting119864(119894) For the sakeof simplicity we will graph 119876119894 versus 119894 and connect all thedifferent values of the electrostatic field in each region bystraight lines An example of a graph is shown in Figure 2

Note that every possible electrostatic field configurationconsists of a closed path of length 2119873 that starts at (0 0) andends at (2119873 0) and is uniquely determined by a sequence ofvertices (1198760 1198761 1198762119873) where 119876119895minus1 is connected to 119876119895 byan edge [10 11] In the mathematical literature this is knownas a closed simple walk of 2119873 steps on the lattice of integerswhere the discrete time that numbers the steps is measured inthe horizontal axis and the positions of the walker on the line

Advances in Mathematical Physics 3

Qi

i

Figure 2 Generalized Dyck paths representing a possible electro-static field configuration of 14 parallel static charged planes

are recorded in the vertical axis Since ourwalks start from theorigin at time 119895 the walker is on site 119904(119895) = 1198760 + 1198761 + sdot sdot sdot +119876119895 A Dyck path is a simple closed walk with the followingconstraint 119904(119895) ge 0 for all 119895 [12]The constraint on aDyck pathmeans that it never falls below the horizontal axis but it maytouch it at any number of integer points Amore general Dyckpath or generalized Dyck paths are simple closed walks thatare allowed to go below the horizontal axis [13] We note then

that all the electrostatic field configurations are the ensemblesof generalized Dyck paths of length 2119873 that start at (0 0) andend at (2119873 0)

Let us denote by Γ the ensemble of generalizedDyck pathsof length 2119873 and consider 120574 isin Γ then the partition functionof the electrostatic field will be given by [14]

Z = sum120574isinΓ

119890minus120573E(120574) (9)

where 120573 = 1119896119861119879 119896119861 = 138 times 10minus23 JK is Boltzmannconstant and119879 is the temperature Equation (9) is the same ascalculating a Feynman amplitude in the frame of generalizedDyck paths that is one sums overall generalized Dyck pathsof length 2119873 and to each whole path is then attributed aweight which is given by the Boltzmann factor

4 Exact Partition Function

In order to calculate the sum of (9) we need to countoverall possible generalized Dyck paths of length 2119873 A verypowerful tool for counting walks is by means of the transfermatrixmethod [15 16] A transfermatrixmethod can be usedto count all possible generalized Dyck paths of length 2119873 Letus introduce a set of (2119873+1) times (2119873+1)matrices of the form119882ℓ which represent the number of simple walks in ℓ stepsSince the paths we are considering are made of steps plusmn1 thenthe transfer matrix for this case is a bidiagonal one given by

119882 =

[[[[[[[[[[[[[[

0 119887minus119873minus119873+1 0 0 0 0119886minus119873+1minus119873 0 119887minus119873+1minus119873+2 0 0 0

0 119886minus119873+2minus119873+1 0 d 0 0 0 d d 119887119873minus1119873minus1 0 0 119886119873minus1119873minus2 0 119887119873minus11198730 sdot sdot sdot 0 0 119886119873119873minus1 0

]]]]]]]]]]]]]]

(10)

where

119886119894+1119894 = 119890minus12057311989422119862119887119894119894+1 = 119890minus120573(119894+1)22119862

for minus 119873 le 119894 le 119873(11)

In this picture the evaluation of (119882ℓ)119894119895 consists in summingthe weights of all walks of ℓ steps from 119894 to 119895 Therefore thepartition function for the random walk of an electrostaticfield generated by means of 2119873 static parallel randomlycharged planes where half of them have a charge surfacedensity 120590 = 1119860 and the other half has a charge surfacedensity 120590 = minus1119860 is given by

Z = [1198822119873 119873 + 1119873 + 1] (12)

where [1198822119873 119873+1119873+1] represents the element in row119873+1and column119873 + 1 of the transfer matrix119882 raised to the 2119873power

For example consider the case of 4 parallel static ran-domly charged planes with surface charge distribution 120590 =plusmn1119860 for this case 119873 = 2 and we can then write down thetransfer matrix for this particular problem which is

119882 =[[[[[[[[[[

0 119890minus1205732119862 0 0 0119890minus41205732119862 0 1 0 00 119890minus1205732119862 0 119890minus1205732119862 00 0 1 0 119890minus412057321198620 0 0 119890minus1205732119862 0

]]]]]]]]]]

(13)


therefore the partition function will be given by

Z = [1198824 3 3] = 2119890minus3120573119862 + 4119890minus120573119862 (14)

Equation (14) is given in the formZ = sum119896 119892(E119896)119890minus120573E119896 where119892(E119896) represents the number of paths that share the sameenergy E119896 this means that for the case when 119879 rarr infin all thepossible electrostatic field configurations will have the sameprobability distribution that is

lim119879rarrinfin

119875 (E119896) = lim119879rarrinfin

119890minus120573E119896Z

= (119873)2(2119873) (15)

and this was the case studied in [8]We can calculate the expectation value for the energy of

the system when 119879 rarr 0 that islim119879rarr0

⟨E⟩ = lim119879rarr0

(minus120597 lnZ120597120573 ) = 1198732119862 (16)

Equation (16) means that for very low temperatures only thepaths with minimum electrostatic energy will contribute tothe energy of the system For this case the system behaveslike119873 parallel plate capacitors connected in series

5 Numerical Analysis

We can perform a direct evaluation of the partition functiongiven in (12) for a given number of charged sheets using thenumerical code given in the appendix By using the numericalcode for 119873 = 1 2 we noticed that the minimum energyof the system is given by Emin = 1198732119862 and the maximumenergy of the system is given by Emax = 119873(1 + 21198732)6119862 Wehave also found the entropic factor for the first twominimumenergies and the maximum energy which invites us to writethe partition function as

Z = 2119873119890minus1198731205732119862 + 2119873minus1 (119873 minus 1) 119890minus(119873+4)1205732119862 + sdot sdot sdot+ 2119890minus119873(1+21198732)1205736119862 (17)

From (17) we see that the paths that contribute the most tothe partition function are given by the first two terms and wealso see that the second term dominates over the first term if

(119873 minus 12 ) 119890minus2120573119862 gt 1 (18)

Equation (18) tells us that when119873 gt 1+ 21198902120573119862 the paths thatcontribute the most to the partition energy belong to the firstexcited state

6 Conclusions

We have shown that it is possible to obtain the exact partitionfunction for the electrostatic field generated by means ofseveral static parallel infinite charged planes in which thesurface charge distribution is not explicitly known We haveworked out the special case where the charged planes have

a constant surface charge distribution given by plusmn1119860 andthe overall electrostatic system is neutral that is there is thesame number of positive and negative charged planesWe usethe connection between generalized Dyck paths and possibleelectrostatic field configurations to obtain the partition func-tion of the system as a discrete Feynman path integral Atransfer matrix approach was used to count overall possiblegeneralized Dyck paths A simple numerical code is given toverify our results

Appendix

In this appendix we present a simple Mathematicacopy codewhich evaluates the partition function given in (12)The onlyvariable that needs to be specified is the number of positiveor negative charged sheets119873

119882 = Table[KroneckerDelta [119894 + 1 119895]

lowast Exp [minus120573 lowast (119894 + 1)22 ]+ KroneckerDelta [119894 119895 + 1]lowast Exp [minus120573 lowast (119895)22 ] 119894 minus119873119873 119895 minus119873119873]

1198822119873 = MatrixPower [119882 2119873] 119885 = Expand [1198822119873 [[119873 + 1119873 + 1]]]

(A1)

Conflicts of Interest

The author declares that they have no conflicts of interest

Acknowledgments

This work was supported by the program ldquoCatedras CONA-CYTrdquo

References

[1] A Lenard ldquoExact statistical mechanics of a one-dimensionalsystem with Coulomb forcesrdquo Journal of Mathematical Physicsvol 2 pp 682ndash693 1961

[2] S F Edwards and A Lenard ldquoExact statistical mechanics of aone-dimensional system with Coulomb forces II The methodof functional integrationrdquo Journal of Mathematical Physics vol3 pp 778ndash792 1962

[3] R J Baxter ldquoStatistical mechanics of a one-dimensionalCoulomb system with a uniform charge backgroundrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol59 no 4 pp 779ndash787 1963

[4] S Prager ldquoThe one-dimensional plasmardquoAdvances in ChemicalPhysics 2007


[5] E H Lieb D C Mattis and F J Dyson Finite MathematicalPhysics in One Dimension Exactly Soluble Models of InteractingParticles Academic Press New York NY USA 1966

[6] V Demery R Monsarrat D S Dean and R Podgornik ldquoPhasediagram of a bulk 1d lattice Coulomb gasrdquo EPL vol 113 no 1Article ID 18008 2016

[7] V Demery D S Dean T C Hammant R R Horgan and RPodgornik ldquoThe one dimensional Coulomb lattice fluid capac-itorrdquo The Journal of Chemical Physics vol 137 no 6 p 0649012012

[8] RAldana J VAlcala andGGonzalez ldquoThe randomwalk of anelectrostatic field using parallel infinite charged planesrdquo RevistaMexicana de Fısica vol 61 no 3 pp 154ndash159 2015

[9] D J Griffiths Introduction to Electrodynamics Prentice-HallUpper Saddle River NJ USA 1999

[10] G Mohanty Lattice Path Counting and Applications AcademicPress New York NY USA 1979

[11] E Deutsch ldquoDyck path enumerationrdquo Discrete Mathematicsvol 204 no 1-3 pp 167ndash202 1999

[12] R Chapman ldquoMoments of Dyck pathsrdquo Discrete Mathematicsvol 204 no 1-3 pp 113ndash117 1999

[13] J Labelle and Y N Yeh ldquoGeneralized Dyck pathsrdquo DiscreteMathematics vol 82 no 1 pp 1ndash6 1990

[14] G M Cicuta M Contedini and L Molinari ldquoEnumerationof simple random walks and tridiagonal matricesrdquo Journal ofPhysics A Mathematical and General vol 35 no 5 pp 1125ndash1146 2002

[15] L Shapiro ldquoRandom walks with absorbing pointsrdquo DiscreteApplied Mathematics The Journal of Combinatorial AlgorithmsInformatics and Computational Sciences vol 39 no 1 pp 57ndash671992

[16] T Jonsson and J F Wheater ldquoArea distribution for directedrandomwalksrdquo Journal of Statistical Physics vol 92 no 3-4 pp713ndash725 1998

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


z

Figure 1 Possible electrostatic field configurations inside a four-charged-plane system where the charge distribution is not known

given to show some features of the partition function In thelast section we summarize our conclusions

2 Model of the System

Suppose that we are given a collection of 2119873 infinite chargedplanes half of them have a constant charge distribution 120590and the other half have a constant charge distribution minus120590that is the system is overall neutral If we randomly place the2119873 infinite charged planes parallel to each other along the 119911-axis at position 119911119899 = 119899119889 where 119899 isin 0 1 2119873 minus 1 thenthe electrostatic field would evolve along the 119911-axis makingrandom jumps each time it crosses an infinite charged sheet

For example consider a system of four charged planesWe know that 119864(119911) equals zero at the left and right side ofthe configuration due to the neutrality of the system [9]After crossing the first charged plane 119864(119911) would increaseor decrease in an amount of 1205901205980 depending on whether thefirst charged plane had a positive or negative surface chargedensity In Figure 1 we show all the possible electrostatic fieldconfigurations for the case when the charge density is notknown

Since we are concern with the statistical properties of theelectrostatic field we have first to obtain the Hamiltonian ofthe field or the electrostatic energy of the system We knowfrom basic electrodynamics courses that the electrostaticenergy is given by [9]

E = 120575119864Δ119881 (2)

where 120575119864 = 120598011986422 is the electrostatic energy density andΔ119881 = 119860119889 is the volume inside the region between twoconsecutive parallel charged sheets Denoting 119864(119894) as thevalue of the electrostatic field in the 119894th region between twocharged sheets placed in 119911 = (119894minus1)119889 and 119911 = 119894119889 then we havethe following boundary condition

119864 (119894) minus 119864 (119894 minus 1) = 1205901205980 119878119894 for 119894 = 1 2 2119873 (3)

where 119878119894 is a random variable which takes one of the twopossible values plusmn1 depending on whether the charged planeat 119911 = (119894 minus 1)119889 is positively or negatively charged

We can write down the boundary condition given in (3)in matrix form in the following way

[[[[[[[[[[

1 0 0 0 0minus1 1 0 0 00 minus1 1 0 0 d d

0 sdot sdot sdot minus1 1

]]]]]]]]]]

[[[[[[[[[[

119864 (0)119864 (1)119864 (2)

119864 (2119873)

]]]]]]]]]]

= 1205901205980[[[[[[[[[[

0119878111987821198782119873

]]]]]]]]]]

(4)

where we have included the initial condition of the electro-static field that is 119864(0) = 0 Solving (4) for the electrostaticfield we have

119864 (119894) = 1205901205980119894sum119895=1

119878119895 for 119894 = 1 2 2119873 (5)

Substituting (5) into (2) we obtain the electrostatic energy ofthe system which is given by

E = (12059802 119860119889sum119894 1198642 (119894)) 120575sum2119873

119894=1119878119894 0 (6)

where the Kronecker delta guarantees the neutrality of thesystem

3 Electrostatic Field and Dyck Paths

In this section we present the relation between the electro-static field and generalizedDyck paths Let us consider for thesake of simplicity that all the charged planes carry a surfacecharge density of 120590 = plusmn1119860 Let 119876119894 be the total amount ofcharge to the left of the 119894th region then we can rewrite theelectrostatic field given in (5) as

119864 (119894) = 1198761198941205980119860 (7)

Note that1198760 = 1198762119873 = 0 and |119876119894 minus119876119894minus1| = 1 The electrostaticenergy in terms of the new variable is given then by

E = 12119862sum119894 1198762119894 (8)

where 119862 = 1205980119860119889It is convenient to investigate the nature of all the possible

electrostatic field configurations by plotting119864(119894) For the sakeof simplicity we will graph 119876119894 versus 119894 and connect all thedifferent values of the electrostatic field in each region bystraight lines An example of a graph is shown in Figure 2

Note that every possible electrostatic field configurationconsists of a closed path of length 2119873 that starts at (0 0) andends at (2119873 0) and is uniquely determined by a sequence ofvertices (1198760 1198761 1198762119873) where 119876119895minus1 is connected to 119876119895 byan edge [10 11] In the mathematical literature this is knownas a closed simple walk of 2119873 steps on the lattice of integerswhere the discrete time that numbers the steps is measured inthe horizontal axis and the positions of the walker on the line


Qi

i





Z = sum120574isinΓ

119890minus120573E(120574) (9)




119882 =

[[[[[[[[[[[[[[



]]]]]]]]]]]]]]

(10)

where

119886119894+1119894 = 119890minus12057311989422119862119887119894119894+1 = 119890minus120573(119894+1)22119862

for minus 119873 le 119894 le 119873(11)


Z = [1198822119873 119873 + 1119873 + 1] (12)



119882 =[[[[[[[[[[


]]]]]]]]]]

(13)



Z = [1198824 3 3] = 2119890minus3120573119862 + 4119890minus120573119862 (14)


lim119879rarrinfin

119875 (E119896) = lim119879rarrinfin

119890minus120573E119896Z

= (119873)2(2119873) (15)




(minus120597 lnZ120597120573 ) = 1198732119862 (16)






(119873 minus 12 ) 119890minus2120573119862 gt 1 (18)


6 Conclusions



Appendix





(A1)



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Qi

i





Z = sum120574isinΓ

119890minus120573E(120574) (9)




119882 =

[[[[[[[[[[[[[[



]]]]]]]]]]]]]]

(10)

where

119886119894+1119894 = 119890minus12057311989422119862119887119894119894+1 = 119890minus120573(119894+1)22119862

for minus 119873 le 119894 le 119873(11)


Z = [1198822119873 119873 + 1119873 + 1] (12)



119882 =[[[[[[[[[[


]]]]]]]]]]

(13)



Z = [1198824 3 3] = 2119890minus3120573119862 + 4119890minus120573119862 (14)


lim119879rarrinfin

119875 (E119896) = lim119879rarrinfin

119890minus120573E119896Z

= (119873)2(2119873) (15)




(minus120597 lnZ120597120573 ) = 1198732119862 (16)






(119873 minus 12 ) 119890minus2120573119862 gt 1 (18)


6 Conclusions



Appendix





(A1)



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Z = [1198824 3 3] = 2119890minus3120573119862 + 4119890minus120573119862 (14)


lim119879rarrinfin

119875 (E119896) = lim119879rarrinfin

119890minus120573E119896Z

= (119873)2(2119873) (15)




(minus120597 lnZ120597120573 ) = 1198732119862 (16)






(119873 minus 12 ) 119890minus2120573119862 gt 1 (18)


6 Conclusions



Appendix





(A1)



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Documents

Exact Partition Function for the Random Walk of an ...downloads.hindawi.com/journals/amp/2017/6970870.pdfExact Partition Function for the Random Walk of an Electrostatic Field GabrielGonzález1,2