9783540729945-c2

2Dynamics of Oscillator Chains

Allan J. Lichtenberg1, Roberto Livi2, Marco Pettini3 and Stefano Ruo4

1 Electrical Engineering and Computer Science Department University ofCalifornia, Berkeley, CA 94720-1770, [email protected]

2 Dipartimento di Fisica and CSDC, Universita` di Firenze via G. Sansone 1, 50019Firenze, [email protected]

3 INAF Osservatorio Astrosico di Arcetri Largo Enrico Fermi 5, 50125 Firenze,[email protected]

4 Dipartimento di Energetica S. Stecco and CSDC, Universita` di Firenze, andINFN, via s. Marta 3, 50139 Firenze, [email protected]

Abstract. The FermiPastaUlam (FPU) nonlinear oscillator chain has proved tobe a seminal system for investigating problems in nonlinear dynamics. First proposedas a nonlinear system to elucidate the foundations of statistical mechanics, the initiallack of conrmation of the researchers expectations eventually led to a number ofprofound insights into the behavior of high-dimensional nonlinear systems. The ini-tial numerical studies, proposed to demonstrate that energy placed in a single modeof the linearized chain would approach equipartition through nonlinear interactions,surprisingly showed recurrences. Although subsequent work showed that the origin ofthe recurrences is nonlinear resonance, the question of lack of equipartition remained.The attempt to understand the regularity bore fruit in a profound development innonlinear dynamics: the birth of soliton theory. A parallel development, related tonumerical observations that, at higher energies, equipartition among modes couldbe approached, was the understanding that the transition with increasing energy isdue to resonance overlap. Further numerical investigations showed that time-scaleswere also important, with a transition between faster and slower evolution. Thiswas explained in terms of mode overlap at higher energy and resonance overlap atlower energy. Numerical limitations to observing a very slow approach to equiparti-tion and the problem of connecting high-dimensional Hamiltonian systems to lowerdimensional studies of Arnold diusion, which indicate transitions from exponen-tially slow diusion along resonances to power-law diusion across resonances, havebeen considered. Most of the work, both numerical and theoretical, started from lowfrequency (long wavelength) initial conditions.

Coincident with developments to understand equipartition was another programto connect a statistical phenomenon to nonlinear dynamics, that of understandingclassical heat conduction. The numerical studies were quite dierent, involving theexcitation of a boundary oscillator with chaotic motion, rather than the excitation of

A.J. Lichtenberg et al.: Dynamics of Oscillator Chains, Lect. Notes Phys. 728, 21121 (2008)

DOI 10.1007/978-3-540-72995-2 2 c Springer-Verlag Berlin Heidelberg 2008

22 A.J. Lichtenberg et al.

the entire chain with regular motion. Although energy transitions are still important,the inability to reproduce exactly the law of classical heat conduction led to concernfor the generiticity of the FPU chain and exploration of other force laws. Importantconcepts of unequal masses, and anti-integrability, i.e. isolation of some oscillators,were considered, as well as separated optical and acoustic modes that could onlycommunicate through very weak interactions. The importance of chains that do notallow nonlinear wave propagation in producing the Fourier heat conduction law isnow recognized.

A more recent development has been the exploration of energy placed on theFPU or related oscillator chains in high-frequency (short wavelength) modes andthe existence of isolated structures (breathers). Breathers are found as solutions topartial dierential equations, analogous to solitons at lower frequency. On oscillatorchains, such as the FPU, energy initially in a single high-frequency mode is found,at higher energies, to self-organize in oscillator space to form compact structures.These structures are chaotic breathers, i.e. not completely stable, and disintegrateon longer time-scales. With the signicant progress in understanding this evolution,we now have a rather complete picture of the nonlinear dynamics of the FPU andrelated oscillator chains, and their relation to a wide range of concepts in nonlineardynamics.

This chapters purpose is to explicate these many concepts. After a historicalperspective the basic chaos theory background is reviewed. Types of oscillators, nu-merical methods, and some analytical results are considered. Numerical results ofstudies of equipartition, both from low-frequency and high-frequency modes, are pre-sented, together with numerical studies of heat conduction. These numerical studiesare related to analytical calculations and estimates of energy transitions and time-scales to equipartition.

2.1 Historical Perspective and Background Theory

2.1.1 Motivation and Counter Intuitive Numerical Results

In the early 1950s, considering what numerical investigations could be per-formed on a rst generation digital computer at Los Alamos National Labo-ratory, Enrico Fermi suggested to Stanislaw Ulam and John Pasta that thefoundations of statistical mechanics could be explored. He proposed using achain of coupled slightly nonlinear oscillators to show that the nonlinearitywould lead to equipartition of energy among the degrees of freedom. Themodel used in the studies was a discretization of a nonlinear spring which toquartic order is given by the normalized Hamiltonian

H =N

i

[p2i2

+(qi+1 qi)2

2+

(qi+1 qi)33

+ (qi+1 qi)4

4

](2.1)

with N unit masses and unit harmonic coupling. The oscillator chain is knownas the FermiPastaUlam (FPU) model. The original simulations were donewith only the term present (FPU- model) or only the term present

2 Dynamics of Oscillator Chains 23

(FPU- model); most subsequent simulations and analysis were done withthe FPU- model. With periodic endpoints the chain is translationally invari-ant, but the original simulations, as with much subsequent work, consideredxed endpoints, related to a physical nite string. Without nonlinear termsthe coordinates can be transformed to uncoupled normal modes, such that theenergy is always conned to the initial modes in which it is placed. For smallnonlinearities (energy in the nonlinear terms small compared to the energy inthe linear terms) it is logical to place the energy in a mode (or modes) of thelinear system (quasi-modes of the nonlinear system) and observe the subse-quent behavior of the mode energies subject to the laws of motion describedby the Hamiltonian in (2.1). The initial numerics, programmed by Mary Tsin-gou for the FPU- chain with xed ends and N1 moving particles, and withall the energy placed in the rst harmonic (k = 1) of the harmonic normalmodes

Qk =

2N

N

i=1

qi sin(

ik

N

)(2.2)

(with N = 32 and = 1/4) gave, for example, the result shown in Fig. 2.1for Q1 = 4 (E = E1 = 0.077). The initial energy was transferred primar-ily into the rst four modes, with an approximate recurrence (within a fewpercent) occurring in a time 1t/2 = 157 fundamental periods. Similar re-sults were obtained for other initial conditions for both the and models,

Fig. 2.1. FPU original mode oscillations


with the results presented in a Los Alamos report in 1955. Unfortunately,the untimely death of Fermi, prevented regular publication until the workappeared in Fermis collected works [1]. The simulations did not answer thequestion of whether equipartition would ultimately be obtained, as was pre-dicted from general dynamical principles of the nonexistence of global isolatingintegrals, see Poincare [2], and the inferences used to support the concept ofergodicity by Fermi himself [3, 4]. Most of the near-term response to the unex-pected result was to try to explain the recurrences. Using perturbative analysisFord [5] and Jackson [6] obtained oscillations of the rst few harmonics, withJacksons approach, using nonlinearly perturbed frequencies, giving results,including the recurrence times, quite similar to the numerical observations.However, the perturbation procedures are nonconvergent, so no conclusionscan be drawn from them about long-time behavior. There has been a sig-nicant body of literature concerned with these recurrences and methods ofanalysis. Generalizations, for example, have considered energy initially in anarbitrary mode (rather than the lowest frequency mode), the major couplingsidentied, and the eect of various numbers of modes used in analyzing thedynamics [7]. It was found, for example, that there is an induction period,i.e. a time during which there is little change in mode energy, if the energyis initially placed in a high-frequency mode, a condition later observed andqualitatively explained (see below).

2.1.2 Chaos Theory: KAM Isolation, Arnold Diusion, LyapunovExponents, KS Entropy

Parallel to the developments, described above, for a high-dimensional Hamilto-nian system, there were developments in low-dimensional Hamiltonian dynam-ics that informed the oscillator-chain results, and ultimately were informed bythose results. In particular, the KAM theorem for coupled degrees of freedom[8, 9, 10] indicated that the generic case was a divided phase space with reg-ular and chaotic orbits interspersed. Numerical observations, in a surface ofsection of a particular two degree of freedom system (the Henon and Heilespotential), indicated mostly regular orbits at low energy, with the chaoticportion of the phase space increasing rather abruptly over a small range ofincreasing energy, until most of the phase space is chaotic [11]. A practicalexplanation of this rather abrupt increase was that local resonances betweenfrequencies of the two freedoms, which modied the structure of the phasespace in their neighborhood, would overlap with increasing energy, producinglarge areas of chaotic motion [12, 13]. For systems with three or more de-grees of freedom KAM surfaces cannot isolate chaotic regions: leading to thepossibility of diusion, in the sense that there are initial data which canreach points in phase space that are arbitrarily far, although such data havea small microcanonical measure when the nonintegrable perturbation is small[14]. Furthermore, a heuristic understanding of a many-dimensional systemwith weak coupling, backed up by simulations (see [15], Sect. 6.5), indicated


the fraction of the phase volume that is stochastic continually increases withincreasing number of freedoms N [16]. Another relevant theoretical result isthat the upper bound on the rate of Arnold diusion is proportional to

exp(

1q

)(2.3)

where is a perturbation parameter [17] and an optimal convergent per-turbation calculation gave a value of q 1/N [18, 19]. This result wouldindicate that the diusion becomes large if 1/N 1, but the result, being ofperturbative type, does not extend to so large a perturbation. Other heuristicforms will be used to estimate the diusion rate in later Sections. Another de-velopment in low-dimensional chaos, that would inform the high-dimensionaloscillator chain research, was the study of the three-particle Toda lattice [20],with Hamiltonian

H =p21 + p

22 + p

23

2+ exp((q1 q3)) + exp((q2 q1)) + exp((q3 q2)) 3

(2.4)which corresponds to three particles moving on a ring with exponentiallydecreasing repulsive forces between them. In addition to the energy, there isa relatively obvious isolating integral, namely the total momentum, reducingthe motion to two degrees of freedom. The HenonHeiles potential [11] is atruncation of the Toda lattice. However, a surface of section of (2.4), calculatednumerically, showed no chaos [21] and Hamiltonian (2.4) was subsequentlyproved to have a third invariant and thus was integrable, i.e. had nonchaoticphase-space trajectories [22].

In order to obtain equipartition it is sucient for the dynamics to be er-godic on the energy shell, i.e. microcanonical averages over a given energysurface and time averages over motions taking place over the same energysurface must be equal. However, since it is physically relevant that the con-vergence to equipartition should occur on a nite time, and be possibly fast,the stronger dynamical property of mixing could be required. A direct nu-merical check of both ergodicity and mixing is impossible in systems withmany degrees of freedom. However, if all trajectories are chaotic and, hence, onthe average exponentially separating, positive KolmogorovSinai (KS) entropy[23, 24] and exponentially fast mixing follow as a consequence. Therefore,an obvious quantity to be examined is the largest Lyapunov exponent, givingthe average separation rate between nearby trajectories as

= limt

1tln(t)(0) (2.5)

where (t) is the tangent vector whose time evolution is described by the tan-gent dynamics equation described in Sect. 2.3. A positive value of indicatesexponential separation of initially close trajectories, i.e. chaos. A diculty ofrealistic Hamiltonian systems is that, in generic conditions, the energy shell


is divided into chaotic and ordered trajectories, and hence mixing can-not occur everywhere in phase-space. Then, the qualitative statement mightbe that, if almost all of the energy surface is characterized by an invariantdistribution that has a positive KS entropy, then, for all practical purposes,equipartition will be reached. The numerical calculation of Lyapunov expo-nents has been used extensively to test for chaotic motion, particularly afterthe numerical techniques were formalized [25]. It was logical that the methodwould be applied to the FPU chain, and became an important element in thenumerical investigations in the 1980s and beyond, as discussed in more detailbelow.

For a more complete introduction to these topics see [15]: Sect. 3.2 (KAMTheory); Sects. 5.2 and 5.3 (concepts of stochasticity) and Sects. 6.1 and 6.2(Arnold diusion). See also [13] and, more rigorously, [26].

2.1.3 Geometrization of Hamiltonian Dynamics

Without attempting to be exhaustive, a few historical comments might behelpful to place the recent contributions about the geometrical approach todynamics which are reviewed in the present Chapter, in a more general con-text.

The idea of looking at the collection of solutions of the Newtons equa-tions of motion from a geometric point of view dates back to Poincare and tothe development of the qualitative theory of dierential equations. Tacklingthe famous problem of the integrability of the three-body problem, Poincarediscovered that generic classical Hamiltonian systems, in spite of their deter-ministic nature, lack predictability because of their extreme sensitivity to theinitial conditions. Such an instability of classical dynamics originates in homo-clinic intersections, which Poincare described in his Methodes Nouvelles de laMecanique Celeste [2] without even attempting to draw them. The methodwas later developed by Cartan among others, using what is now called sym-plectic geometry [27]. Although of undeniable elegance, symplectic geometryis not very helpful to advance our knowledge about the regions in phase spacewhere the dynamics is unstable. The name of Poincare, together with that ofFermi, is also associated with an important theorem about the nonexistenceof analytic integrals of motion, besides energy, for generic nonlinear Hamil-tonian systems describing at least three interacting bodies [3, 4]; this is theorigin of the concept of topological accessibility of the whole constant energyhypersurface of phase space with high degree of freedom systems, with genericinitial conditions.

In the 1940s, a qualitatively new attempt was made to make use of geomet-ric concepts to relate Newtonian dynamics with statistical mechanics. Krylov[28] showed in for the rst time the existence of a relationship between dy-namical instability (seen as the exponential amplication of small deviationsin the initial conditions of a collection of colliding objects representing ide-alized atoms in a gas) and phase space mixing. Phase mixing is a stronger


property than ergodicity and is far more relevant to physics than ergodicity.In fact, while ergodicity assures the equality of time and phase space averagesof physical quantities, phase mixing addresses the rate of approach to ensem-ble averages in a nite time. In modern terms, Krylov realized the necessityof chaotic dynamics to obtain fast phase mixing for the physically relevantobservables and to make the connection between dynamics and statisticalmechanics stronger. But Krylov also has the great historical merit of havingattempted to bridge the dynamical foundations of statistical mechanics with apowerful eld of mathematics, Riemannian dierential geometry. Krylov knewmathematical results, concerning the properties of geodesic ows on compactnegatively curved manifolds, by Hadamard [29], Hedlund [30] and Hopf [31].He envisioned their potential interest to physics, once Newtonian dynamicsis rephrased in terms of Riemannian geometric language. Such a possibilitywas well known since the beginning of the century, mainly due to the workof Levi-Civita; in particular that the principle of stationary action entails theclose connection of a classical mechanical ow with a geodesic ow in a con-guration space endowed with a suitable metric. Krylovs eorts concentratedon the analysis of the properties of physical systems which move in negativelycurved regions in conguration space. For example, he discussed how the pres-ence of an inection point in the Lennard-Jones potential could inuence thedynamics of a dilute gas (through the appearance of regions of negative scalarcurvature in conguration space) and its ensuing strong instability. These at-tempts have been very inuential on the development of the so-called abstractergodic theory, where Anosov ows [32] (e.g., geodesic ows on compact man-ifolds with negative curvature) play a prominent role. Ergodicity and mixingof these ows have been thoroughly investigated. To give an example, Sinaiproved ergodicity and mixing for two hard spheres by just showing that sucha system is similar enough to a geodesic ow on a negatively curved compactmanifold [33]. Krylovs intuitions have been worked out further by severalphysicists amongst whom we cite those of [34, 35, 36, 37, 38, 39, 40]. Theydiscovered, much to their surprise, that geodesic ows associated with phys-ical Hamiltonians do not live on negatively curved manifolds, despite theirchaoticity. Only a few exceptions are known, in particular two low-dimensionalmodels [35, 36, 41], where chaos is actually associated with hyperbolicity dueto everywhere negatively curved manifolds. In fact, for certain models the re-gions of negative curvature of the mechanical manifolds apparently shrink byincreasing the number N of degrees of freedom, thus reducing the frequencyof the visits of negatively curved regions.

This somewhat biased search for negative curvature has been the mainobstacle to an eective use of the geometric framework originated by Krylovto explain the source of chaos in Hamiltonian systems. On the other hand, it istrue that the Jacobi equation, which describes the stability of a geodesic ow,is in practice only tractable on negatively curved manifolds. Formidable math-ematical diculties are encountered in treating the (in)stability of geodesicows on manifolds of nonconstant and not everywhere negative curvature.


Moreover, for this kind of problem, intuition can hardly help. However, theadvent of computers has been of invaluable help. As a matter of fact, dur-ing the last few years an interplay between analytic methods and numericalsimulation has made it possible to overcome the diculties, showing that theRiemannian geometric approach can be applied to dynamical systems of inter-est to statistical mechanics, eld theory, and condensed matter physics [42].This has extended the domain of application of geometric techniques, and hasalso introduced a new point of view about the origin of chaos in Hamiltoniansystems, as well as new methods to describe and understand it, new in asense that will be made clear in Sect. 2.4.7.

A more detailed exposition of the geometric method and its application tocalculating Lyapunov exponents, which we will be summarizing in Sects. 2.4.7and 2.8.2, can be found in [42]. See also mathematical expositions in [27, 28].

2.1.4 Development of Soliton Theory

It is somewhat ironical that the most celebrated result that came out of theinvestigation of the FPU chain did little to resolve Fermis original questionof whether or not the nonlinearity would lead to equipartition among thedegrees of freedom. In an attempt to understand the apparent stability ofthe recurrences Norman Zabusky and Martin Kruskal [43, 44] found a Taylorexpansion of the discreteness, valid for long wavelength modes, that recoveredpartial dierential equations, dierent from the original nonlinear spring whichproduced the discretized chain of oscillators. The resulting equations are theKortevegde Vries (KdV) equation for the FPU- chain and the modiedKortevegde Vries (mKdV) equation for the FPU- chain [43, 44]. The latterchain, with appropriate normalizations, gave the standard form

u + 12u2u + u = 0 (2.6)

where = h3t/24, = x ht, h = L/N , L with the length of the string andN the number of oscillators. Nonlinear equations of this and related typeshad been known to have stable traveling solutions, where the dispersion andnonlinearity balance to produce constant amplitude and propagation velocity.An arbitrary initial condition, such as the lowest linear mode on the FPU-chain, breaks up initially into a set of structures each having a steady travel-ing solution with its own velocity. Remarkably, these structures are sucientlystable that they pass through one another without breaking up, and the ob-served recurrences can be interpreted in terms of their superpositions. Butthese results do not improve on the best perturbation calculations, and areclearly limited to long-wavelength (low-frequency) modes by the approxima-tions which led to (2.6). Partial dierential equations, like (2.6), have an in-nite number of freedoms, such that general integrability from arbitrary initialconditions requires an innite number of invariants of the motion. The realexcitement came when it was shown that such an innite set exists for (2.6),


and the new eld of soliton theory and applications was born, which wouldtake us far from the subject at hand. A nal note, which is important to ouroverall understanding, is that a single initial nonlinear mode solution of themKdV equation was found to become unstable as the energy is increased. Alinearization around the nonlinear structure predicted the unstable wave num-bers and growth rates, and showed that the values correspond to the observedmode growth for the same discretized structure on the FPU- oscillator chain[45, 46]. The result in which one soliton decomposes into a nite number isnot inconsistent with general soliton theory. The instability will give us insightinto some later results.

2.1.5 Resonance Overlap Explanations

Using the concept of mode overlap to estimate the transition between regularand chaotic motion Felix Izrailev and Boris Chirikov obtained estimates formode overlap both for low- and high-frequency modes [47]. Although there arevarious approximations required to obtain results, a simple numerical estimatecan be made by equating the nonlinear frequency shift k to the modespacing k, i.e. setting k/k 1. The mode overlap estimate from thisapproximation, in terms of energy density, is

k =EkN

={

4/(3k) , k N2k2(N k)/(3N2) , (N k) N . (2.7)

The result for long-wavelength modes, k N , is not a necessary condition,as seen in many subsequent numerical experiments, but approximates anothertransition, discussed below, between weak and strong stochasticity (the SST).The result for short-wavelength modes, (Nk) N , is neither necessary norsucient. It predicts easy overlap at short wavelengths due to mode crowding,while numerical simulations show consistently that, from a practical point ofview, equipartition is more readily obtained from long-wavelength than fromshort-wavelength initial conditions. General theoretical arguments as to theaccessibility of modes has been advanced to show that this is the case [48, 49].

We discuss a resonance overlap criterion, as presently used, in Sect. 2.4.1.The concept, initially proposed by Chirikov for two degrees of freedom andreviewed by him, including higher dimensionality, [13], can also be found in[15], Chap. 4.

2.1.6 Numerical Methods

The straightforward method of computing Lyapunov exponents, using (2.5),particularly the largest exponent, was a powerful numerical tool for statisti-cally investigating the dynamical properties of oscillator chains. Another veryuseful statistical quantity is the information entropy


S =

k

ek ln ek (2.8)

with ek = Ek/

k Ek, such that S = 0 if all the energy resides in a singlemode, and has a maximum S = lnN if the energy is uniformly distributedamong all modes. By using (2.5) and (2.8), detailed numerical investigationswere carried out among investigators in Florence [50, 51, 52, 53], startingfrom long-wavelength modes of the FPU- system, obtaining the variationof and S with energy density = E/N . They found a distinct break inthe behavior between weak stochasticity at lower values of , having strongpower-law dependencies of and S on , and strong stochasticity at higher with weak dependence. The transition (SST) is qualitatively related to themode overlap criterion (2.7). Note that is not necessarily a small quantity.

2.1.7 Methods of Analysis and Numerical Results

It is clear from the phase space description of high-dimensional systems thatmode overlap is not necessary to obtain positive Lyapunov exponents. Mostgeneric initial conditions will lie in stochastic layers, exhibiting > 0. Thequestion becomes what determines the rate of energy diusion between the de-grees of freedom? One approach to this problem is to isolate a few of the mostclosely coupled modes and determine if their resonant interaction results inchaos that can then couple to other resonances. This was done, as describedpreviously, for low-dimensional chaos to understand the exponentially slowArnold diusion. For high-dimensional systems the situation is more com-plicated with rapid diusion across overlapped resonances and slow Arnolddiusion along resonance (see [15], Chap. 6, for a detailed discussion). Themethod of isolating a few interacting resonances and then calculating theircoupling to the larger phase space was used for another oscillator chain, adiscretized sine-Gordon equation, to explore the transition from power-law(numerically observable) equipartition rates with varying , to exponentiallyslow (not numerically observable) rates [54]. It was also found, using thisapproach and comparison with numerics, that short-wavelength mode inter-actions required considerably higher energy to produce chaos. The methodwas then applied to the FPU- chain, in more detail, specically investigat-ing the process by which stochastic interaction between a few long-wavelengthmodes was transferred to short-wavelength modes and calculating a transitionbetween exponentially slow and power-law scaling of the energy transfer [55].At about the same time, there was considerable attention given to determin-ing the scaling of the equipartition time Teq with , in the power-law regime,nding Teq 3 [56, 57, 58] with the latter references giving a heuristic cal-culation of this scaling. The numerics and method of estimation will be givenin Sects. 2.5 and 2.8, respectively. Other authors have tted the data to astretched exponential, Teq exp(1/4) [59], obtaining a better agreementover a wider range, but without any theoretical underpinning. Indeed, the


reason for this scaling has not been explicitly explained, nor has its relationto the power-law scaling. It is very likely that, as is decreased, longer andlonger time-scales come into play and, therefore, no denite functional formwill be able to t the increase of the time-scale over the full small range.The scaling 2, detected at smaller than the threshold value of the SSTtransition ( 1), has also not been specically related to the Teq 3scaling in the same range. The scaling at higher , that is at larger thanthe threshold value of the SST transition, has been heuristically determinedusing a random matrix approximation for the tangent dynamics, intuitivelysuggesting that above the SST chaos is fully developed [53]. The scalings of with can now be determined, analytically, by considering the geometry ofthe phase space near equipartition. Making suitable assumptions about thegeometry of mechanical manifolds, the scaling of with , both below andabove the SST transition and the value of at the transition has been theoret-ically calculated in agreement with numerical ndings [42, 60, 61]. Althoughthe method was developed to understand the FPU- scaling, it is applicableto oscillator chains with various force laws, as can be found in the referencedworks. The mathematical procedure is outlined in Sects. 2.4.7 and 2.8.2.

2.1.8 Comparison of Dierent Oscillator Chains

Although the FPU- oscillator chain has received most of the attention, therehas been, from the beginning, interest in other force laws. The cubic potentialin the FPU- model is more conducive to using expansion procedures to obtainanalytic estimates [62], and also, the form with periodic boundary conditionsand appropriately chosen is a third order truncation of the Toda lattice po-tential, which is integrable. However, the FPU- is not energy renormalizablewith varying (does not scale with E), and furthermore suers from theproblem of unbounded trajectories at high energy. Nevertheless, comparisonwith the FPU- dynamics has added considerably to our overall understand-ing. If the nite time version of the Lyapunov exponent (2.5) is calculated forthe N -particle Toda lattice and its FPU- approximation, the two exponentsdecrease, without separation, until some induction time or trapping timeT(), is reached, after which FPU attains a constant value, while Todacontinues to decay, as it must for an integrable system [61]. Plotting T(),with N as a parameter, in the weak stochasticity regime, it was found in [61]that T 2, which is dierent from the Teq 3 scaling found both for theFPU- and the FPU- systems, i.e. the trapping time and the equipartitiontime scale dierently with . A transition at some small to a rapid increasein T with decreasing , with the transition value a function of N , was alsoobserved and interpreted as a transition to regular motion. This phenomenonhad been observed earlier in the discretized sine-Gordon system and inter-preted in a similar fashion [54]. However, subsequent work with the FPU-system elicited a dierent interpretation, that the transition was to the ex-ponentially slow form of Arnold diusion [55]. These dierent interpretations


have not been theoretically reconciled. The Hamiltonian containing both cubicand quartic nonlinearities, as in (2.1), has also been investigated [63].

The contrast of the FPU oscillator chain with other types of chains has alsoled to considerable insight and some additional puzzles. The class of KleinGordon chains, with on-site potentials, are similar to the FPU, but morecomplicated, both because they lack the FPU translational invariance andbecause they have an additional parameter whose scaling must be determined.In addition to the sine-Gordon version, a closer comparison with the FPU-chain employs a KleinGordon on-site potential having quadratic and quarticterms, with Hamiltonian

H =N

i=1

p2i2

+(qi+1 qi)2

2+

mq2i2

+q4i4

, (2.9)

which is often called the 4 model to distinguish the quartic nonlinearityfrom other KleinGordon potentials. In the rst comprehensive numericalcomparison of the two systems, (2.1) and (2.9), some physical dierences wereobserved and, qualitatively, understood. In particular, at a given , the 4 tooksignicantly shorter time to obtain equipartition from long-wavelength modeinitial conditions and signicantly longer time from short-wavelength modes,than the FPU [64]. The fact that from short wavelengths it was generally alonger process to obtain equipartition was remarked in that early work, butlittle background theory had been done for these initial conditions. A latercomparison for the long wavelengths provided a more complete numericalstudy and was able to explain quantitatively these dierences [65]. We willpresent the numerical comparisons in Sect. 2.5 and outline the supportingtheory in Sect. 2.8. The understanding of the results from short wavelengthsawaited the development of new theoretical concepts, as given below. Beforeconsidering this subject we note that the emphasis on energy density, holding constant as N is varied, i.e. E N , is not always the relevant way to look ata problem, as seen in calculating resonance overlap [55]. The case in which Eis held xed as N is varied has been used to analytically calculate stochasticitythresholds of the FPU and 4 models [66].

2.1.9 Dynamics at Short Wavelengths: Chaotic Breathers

Following the original numerical work, most numerical studies examined theevolution from long-wavelength (low-frequency) modes in which neighboringoscillators are nearly in phase. Zabusky and Deem [67] were the rst to con-sider the case in which the energy is put into a high-frequency mode. In theirearly work, the zoneboundary mode was excited with an added spatial modu-lation for the FPU- model. Our main concern here will be the FPU- model,and spatial modulation of the mode is spontaneously created by modulationalinstability. Budinsky and Bountis [68] found that the zoneboundary -mode,


i.e. the mode with 180 phase shift between neighboring oscillators of the one-dimensional FPU lattice is unstable above a given energy threshold Ec whichscales like 1/N . This result was later conrmed by Flach [69] and Poggi andRuo [70], who also obtained the exact numerical factor relating Ec to 1/N .These results were obtained using a direct linear stability analysis around theperiodic orbit corresponding to the -mode. Similar methods have been morerecently applied to other modes and other FPU potentials by Chechin [71, 72]and Rink [73]. A technique which allows for a more general exploration of thedynamics starting from short wavelengths is to follow an envelope function ofthe oscillators dened by i = (1)iqi. Since the main phase variation of theoscillator amplitudes qi vary by nearly from one oscillator to the next, the ivary slowly; a Taylor expansion of the envelope function in the oscillator spacecan produce a dierential equation whose equilibrium properties, stability, andnonlinear eects can be explored (see Sect. 2.4.5). A formula for Ec, valid forall N , has been obtained in Refs. [74, 75, 76, 77] in the rotating wave ap-proximation (RWA) given in (2.86). Besides calculating the energy threshold,the growth rates of mode amplitudes were obtained. The application to theKlein-Gordon lattices was rst studied by Kivshar and Peyrard [78], followingan analogy with the BenjaminFeir instability in uid mechanics [79]. A dif-ferent approach to describe this instability had been previously introduced byZakharov and Shabat [80], who studied the associated nonlinear Schrodingerequation in the continuum limit. Using that method for the FPU equations ofmotion, the instability boundary was found by Berman and Kolovskii [81] inthe so-called narrow-packet approximation. Detailed numerics over longertimes were obtained for the FPU model by Pettini [64] and for the discretizedsine-Gordon equation by Goedde [54], both indicating that, for a given en-ergy, short-wavelength (high-frequency) modes required longer times to reachequipartition than long-wavelength modes. At about the same time it wasdemonstrated that stable intrinsic localized modes (ILMs) could exist for an-harmonic periodic structures [82]. However, from more general high-frequencyinitial conditions there was a tendency to form ILMs but they were not stable,breaking up and ultimately decaying toward equipartition [75, 83].

The existence of ILMs (also called breathers) on periodic chains and thecomplex behavior of more arbitrary high-frequency initial conditions has ledto extensive study of these structures to understand their stability. A com-prehensive review of these studies would lead us far from the main topic ofthis review (see [84] for a review and further references). The breathers canbe stationary or moving, and, like low-frequency solitons, can pass throughone another. Whether energy is exchanged in such an interaction depends onthe systems stability properties. ILMs that are not exact solutions of the un-derlying system generally exchange energy, and in a particular process havebeen shown to transfer energy from the smaller to the larger breather [85].This phenomenon is also observed numerically for a KleinGordon chain [86]and for the FPU- chain [87, 88]. For xed end-points, as in the original FPUstudies and much subsequent work, a clearly dened instability boundary


cannot be calculated for a discrete chain. Nevertheless, as described below,some approximate results are available.

The four-mode resonance overlap criterion for a stochasticity transition,described in Sect. 2.4.1 for low-frequency modes, has also been used for high-frequency modes for the discretized sine-Gordon chain [54]. It predicted theincreased stability for these modes, as found numerically. Another approach toa reduced problem is to represent the main-energy containing oscillator andthe immediate neighboring oscillators as a three degree of freedom systemfrom which a mapping can be obtained [89]. In this reduced phase space, de-pending on the energy and the action, one observes both regular and chaoticregions. The chaotic regions are sucient to indicate chaos in the larger sys-tem, but do not give a time-scale for equipartition to be approached. Theregular regions may also be chaotic in the larger system, but are generallymore weakly so. Note that the mapping presentation uses initial conditionsclose to those of a breather, which is narrow in oscillator space, and there-fore has a broad distribution of energies in mode space. Contrarily, the modepresentation starts with a narrow distribution in mode space and thereforea broad distribution in oscillator space. The technique employing the enve-lope equations and the RWA has been used to describe longer time eects,as well as instability boundaries both for periodic boundary conditions at lowenergies [90, 91] and xed boundary conditions at both low and high energies[92, 93, 94], and used to compare the dynamics of the FPU and 4 chains [94].

The general picture that has emerged is that if the energy is placed ina high-frequency mode or modes for which neighboring oscillators are pri-marily out of phase, a complicated dynamics ensues, which consists of threestages. First, there is an initial stage in which, for suciently high energy,the mode breaks up into a number of breather-like structures. Second, on aslower time-scale, these structures coalesce into one large unstable structure,called a chaotic breather (CB). Since a single large CB closely approximatesa stable breather, a third and nal decay stage, toward equipartition, can bevery slow. One does not know whether there exists any true energy thresholdto achieve equipartition, although there appears to be some numerical evi-dence for such a threshold in the discretized sine-Gordon system. However, asdiscussed extensively with respect to low-frequency mode initial conditions,the practical thresholds refer to observable timescales.

For nonlinear structures on chains having weak spring potentials, forwhich the nonlinear restoring force substracts from the linear restoring, theinteraction that causes the nal decay is radiation from the breathers to thepropagating linear modes. For strong spring potentials the breather fre-quency is above the optical band, so a more subtle energy interchange mustoccur [84, 87, 95, 96]. A beat phenomenon has been postulated as the energyinterchange mechanism, and used to calculate an -scaling that agrees withnumerics [92].


2.1.10 Heat Transport in Lattice Models

A main goal of classical kinetic theory is to provide the denition of trans-port coecients through phenomenological constitutive equations. The ba-sic hypotheses of this macroscopic theory of transport phenomena are theassumption that uxes are proportional to thermodynamic forces and thatthe system evolves close to equilibrium [97]. For instance, when dealing withheat transport in a solid, one denes the thermal conductivity through theFourier law

J = T , (2.10)where the heat ux J is the amount of heat transported through the unit sur-face in unit time and T (x, t) is the local temperature. Such a phenomenologicalrelation was rst proposed in 1808 by Fourier as an attempt to explain thephenomenon of the Earth cooling. Equation (2.10) is assumed to be valid closeto equilibrium. Actually, the very denition of the local energy ux J(x, t) andtemperature eld T (x, t) relies, in turn, on the local equilibrium hypothesis,i.e. on the possibility of dening a local temperature for a macroscopicallysmall but microscopically large volume in position x at time t.

The rst and most elementary attempt to give a microscopic foundation toFouriers law dates back to Debye [98]. By rephrasing the results of the kinetictheory for the (dilute) phonon gas, he found that the thermal conductivityshould be proportional to Cv, where C is the heat capacity and v, arethe phonon mean velocity and free path, respectively. Moreover, Debye alsorealized that at a microscopic level the nite thermal conductivity in crystalsshould be a consequence of the nonlinear forces acting among the constituentatoms [98].

Peierls further extended the conjecture of Debye and formulated aBoltzmann-like equation, which shows that anharmonicity is necessary forobtaining genuine diusion of the energy by the so-called Umklapp processes,where the nonlinearity is introduced phenomenologically in the transportequation, independently of the microscopic nature of the interactions [99].Nonetheless, the BoltzmannPeierls approach represented an improvementin the theory of lattice thermal conductivity. It allows one to compute thedependence of on the temperature which agrees reasonably well with exper-imental data in the very low-temperature regime. However, basic questionsremained, such as under which conditions is local equilibrium obtained in aphysically accessible time? This kind of a problem partly inspired the nu-merical experiment by Fermi, Pasta and Ulam, as Fermi was aware of theconceptual diculties concerning the possibility of constructing a satisfactorymicroscopic approach to transport theory. In nonlinear chains, the complexinteractions among the constituent atoms or molecules of a real solid are re-duced to harmonic and nonlinear springs, acting between nearest-neighborequal-mass particles. Despite such simplications, the basic ingredients thatone reasonably conjectures to be responsible for the main physical eect (i.e.the niteness of thermal conductivity) are contained in the model. As already


described, the original study expected to verify a common belief, which hadnever been put to a rigorous test: an isolated mechanical system with many de-grees of freedom should eventually yield equilibrium through thermalizationof the energy. Furthermore, the measurement of the time interval needed toapproach the equilibrium state, i.e. the relaxation time of the chain of oscil-lators, would have provided an indirect determination of thermal conductivity, since Debyes argument predicts Cv/r, i.e. inversely proportional tothe relaxation time r /v, which is assumed to represent the average timeneeded for a phononic excitation to relax to thermal equilibrium.

After the lack of success of the FPU numerical experiment, the rst im-portant attempt to reconsider the problem of heat transport in solids from atheoretical point of view was to consider a homogeneous harmonic chain withxed boundary conditions in contact with stochastic Langevin heat baths[100]. The equations of motion

qn = 2(qn+1 2qn + qn1) + n1(+ q1) + nN ( qN ), (2.11)where are independent stochastic processes with zero mean and variance2kBT, with T+ > T, can be solved by a phase-space description, i.e.using the FokkerPlanck equation. However, the solutions were not success-ful in reproducing the Fourier law. They predicted that the heat ux wasproportional to the temperature dierence, rather than the temperature gra-dient, thus showing that homogeneous harmonic chains do not exhibit normaltransport properties. Although there are many aspects of linear chains, such asthe inclusion of disorder or varying masses [101] that we have not consideredabove, our main concern here is with nonlinear chains. Numerical studies ofheat conductivity in the FPU chain were reconsidered at the end of the 1960s.In particular, nonequilibrium simulations of the FPU model (2.1) with cou-pling constants and xed to represent the leading terms of the expansionof the Lennard-Jones potential were performed [102, 103]. These authors alsoconsidered the eect of disorder by including in the model either a disorderedbinary mixture of masses [102] or random nonlinear coupling constants [103].The combination of nonlinearity and disorder did not help the researchers toobtain a clear understanding of the problem. They even found cases in whichanharmonicity increases thermal conductivity. The attention was mainly fo-cused on the form of the temperaure prole T (x). They noticed that its shapedepended on the existence of disorder. Although it is known that T (x) is not aself-averaging observable for disordered harmonic chains, it is not known howT (x) depends on disorder over long enough time-scales in anharmonic chains.Additional questions that were investigated concerned the concentration ofimpurities [102].

Preliminary work on homogeneous anharmonic chains considered theequal-masses FPU and Lennard-Jones chains composed of 30 particles andcoupled with Langevin baths at their boundaries [104], a task that was unfea-sible with the computer resources available at that time. As a consequence,several attempts of designing easy-to-simulate toy models followed these rst


studies. Some examples are reviewed in [105]. One was the so-called ding-a-ling model, a prototype of all models with an on-site potential, as describedin (2.16) in the next Section. This model was found to exhibit normal ther-mal conductivity. The increase in computer power led to a revival of theheat conduction problem inbetween the mid-1980s and the mid-1990s, whennonequilibrium simulations of the FPU model [106, 107] and of the diatomicToda chain [108, 109, 110, 111] of alternating light and heavy masses were per-formed. Subsenquently, there were systematic studies on the size dependenceof the heat conductivity for the FPU chain with quartic [112, 113, 114] or cubic[115] nonlinear potential as well as for the diatomic Toda chain [116, 117].They indicated a divergence of the heat conductivity with N , the numberof mass points, which was interpreted as due to ballistic transport of energythrough the chain. As we will comment in the following, an on-site potentialdetermines a classical conductivity.

2.2 Formulations: Types of Oscillator Chains

2.2.1 Chains Similar to the FPU

Over the years, since the rst numerical investigation by Fermi, Pasta andUlam, many dierent oscillator chains have been studied. There have been var-ious reasons for the particular choices, sometimes because they approximatedphysical systems, sometimes for their simplicity, and sometimes designed tobring out specic features or compare results with other chains.

In choosing an oscillation chain for the initial study, the FPU- system wasa reasonable choice, as it is a discretization of the partial dierential equationfor the nonlinear string with a strong nonlinear restoring force

2y

t2

2y

x2

[1 + 3

(y

x

)2]= 0. (2.12)

The discretization of y(x, t) as yj(t),

y

x=

yj+1 yjx

oryj yj1

x, (2.13)

2y

x2=

yj+1 2yj + yj1x2

, (2.14)

where x = L/N , with L as the length of the string and N 1 the numberof oscillators, yields

yj =(yj+1 2yj + yj1)

x2

{(1 + )

[(yj+1 yj)2

x2+

(yj yj1)2x2

+(yj+1 yj)(yj yj1)

x2

]}

(2.15)


The original work, and also the analytic investigation by Izrailev and Chirikov[47], were with xed end points at j = 0 corresponding to x = 0, and j = Ncorresponding to x = L, such that j = 1, 2 . . . , N1 for the moving oscillators.The coordinates can be rescaled at xed N to any x and L to give theFPU- part of (2.1), with x normalized to 1. Letting x x = L/Land introducing the change of variables yj yjL/L and t tL/L leadsto (2.15) again. Since dyj/dt

= dyj/dt, the energy per mode is unchanged.Thus increasing N by adding oscillators to the end of the chain at xed x isequivalent to adding oscillators by subdividing the chain at xed L, providedthe time and displacement are rescaled.

The addition of the term to (2.1) is a logical extension to a more generalrestoring force. However the and terms have dierent properties, with theenergy scaling dierently with choices of and , such that the energy E isrenormalizable with E and with 2E. Furthermore, the sign in the nonlinearterm in (2.12) changes the behavior from a strong to a weak spring, while the term has a directional antisymmetry. Some of the consequences of thesedierences will emerge in the following sections.

It also became clear in subsequent years that, while xed endpoints werea physical condition for an actual string, periodically continued endpoints (ormass points on a circle) had some attractive features for analysis. With a pe-riodic boundary condition (BC), waves traveling in a single direction withoutreection are allowed, which is a key ingredient in the development of solitontheory, as we outline in Sect. 2.4.3. For a periodic BC, linear momentum is anexact invariant, which simplies various analyses. If the oscillator dynamics isexpressed in terms of linear modes, i.e. the modes which would be exact so-lutions in the absence of the nonlinearity, other dierences between xed andperiodic boundaries become evident. For a periodic BC there is a highest fre-quency boundary mode that has exact alternation of oscillator phases, whichis an exact solution of the nonlinear problem, as considered in subsequentsections.

The FPU type of oscillator chains did not realistically represent the dy-namics of solid materials. A more general representation is given by the Hamil-tonian

H =

i

[p2i2m

+ U(qi) + V (qi+1 qi)]

(2.16)

where U and V are on-site and inter-site potentials, respectively, which aremost generally nonlinear. They can be constructed as physical models of one-dimensional crystals or by discretizations of KleinGordon partial dierentialequations. For the FPU chain U = 0. One form of (2.16) that is used tocompare to FPU- dynamics is the 4 chain with V = (1/2)(qi+1 qi)2 andU = (m2/2)q2i + (/4)q

4i , as given in (2.9), and compared theoretically and

numerically with the FPU in various subsequent sections. The Hamiltonianof (2.16) and the simplied form (2.9) are not rescalable as is the FPU-, butthe coecients can be chosen to make useful comparisons.


An interesting special case of the KleinGordon class of partial dierentialequations is the sine-Gordon equation

yt yxx + sin y = 0, (2.17)which can discretized in space in the same manner as the nonlinear spring toobtain the system Hamiltonian

H =N

i=1

12p2i +

N

i=1

(1 cos yi) +N

i,j=1

Aijyiyj , (2.18)

where the coupling matrix Aij is given by

Aij =(2ij i,j1 i,j+1)

(x)2, (2.19)

pi = y, x = L/N , and ij is the Kronecker . As with the more generalforms of the discretized KleinGordon, (2.18) is not rescalable on x, suchthat both L and N enter as essential parameters. The discretized system is ofparticular interest, as the partial dierential equation is integrable, unlike thenonlinear spring, so the discretization, itself, becomes the only source of chaos.However, at low frequencies (long wavelengths) where the FPU approximatesan integrable system, the transitions are similar [54]. One interesting feature inthat work was an explicit discretization of time, so forming a 2N -dimensionalsymplectic map to be analyzed. The sine-Gordon on-site potential has alsobeen used as an interaction potential to study the Fourier heat law, as wediscuss in Sect. 2.7.2. For that case the potential is known as the FrenkelKontorova potential.

Closely related to the FPU chain is the chain with the same interparti-cle potential structure but with varying masses. To explore the question ofwhether a fraction of the modes, in a distinguishable mode packet, could beisolated from the modes initially containing the energy, Galgani et al. [118]considered a modied FPU- model, with xed ends, described by the Hamil-tonian

H =N

i

[p2i2mi

+(qi+1 qi)2

2+

(qi+1 qi)44

](2.20)

with mi = 1 for i odd and mi = m < 1 for i even. The alternation ofmasses separated the linear mode spectrum into branches, an acoustic branchwhich is only slightly modied from the usual spectrum, and an optical branchassociated with the lower mass particles. The form of the spectrum is givenin Sect. 2.2.2 and the implication for mode isolation in the thermodynamiclimit is discussed in Sect. 2.4.2

Another oscillator chain of particular importance is the Toda lattice, whichgeneralizes the three-particle lattice, given in (2.4), to N particles. The lat-tice, with exponential forces between particles, is generally thought of as con-strained on a ring, which is equivalent to periodic BC. This discretized chain


is exactly integrable, which makes it uninteresting in itself, but very usefulfor comparing to the FPU- potential that can be considered to be a trun-cation of the Toda potential (see Sect. 2.4.3). The same comparison of thethree particle Toda potential with its truncation, the Henon and Heiles po-tential [11], was very useful in understanding low-dimentional chaos, as wehave already discussed briey in Sect. 2.1.2. The N -particle Toda chain hasalso been used to explore the eect of alternating heavy and light masses,which is not integrable, and is discussed briey in Sect. 2.7.1 in connectionwith heat conduction.

The oscillator chains described above do not exhaust the useful types thathave been explored in a variety of contexts. One such chain of historical impor-tance is the ding-a-ling model consisting of alternately harmonically boundand free hard-core particles, which was used to obtain the Fourier law of heatconduction [119]. It is given in (2.132), Sect. 2.7.2, and its properties are dis-cussed there. Variants of the ding-a-ling model have also been studied in thiscontext [120]. The less articial potentials of the KleinGordon type also canproduce the Fourier law, and are considered in Sect. 2.7.2.

2.2.2 Representation in Modes of the Linear System

We have already mentioned that, for linear chains, the transformation to theharmonic normal modes, as given by (2.2), gives a set of mode amplitudesQk that are invariant under the motion. This can be seen by applying theinverse transform, with N moving particles, to the FPU- chain (2.1) or the4 chain (2.9)

qi =

2

N + 1

N

k=1

Qk sin(

ikN + 1

), (2.21)

to obtain [121]

H =N

k=1

12(P 2k +

2kQ

2k

)+

8N + 8

N

i,j,k,l=1

C(i, j, k, l)QiQjQkQl, (2.22)

where Pk are the corresponding momenta, with the mode frequencies kgiven by

k = 2 sin(

k

2N + 2

)(2.23)

for the FPU-, and

k =

m2 + 4 sin2(

k

2N + 2

)(2.24)

for the 4. It is immediately apparent from Hamiltons equations that thedynamics of the linear modes are independent of one another. The quarticterms couple the modes together, with


C(i, j, k, l) = ijkl

P

B(i + j + k + l) (2.25)

for the FPU-, and, for the 4,

C(i, j, k, l) =

P

B(i + j + k + l) . (2.26)

The sum is over the eight permutations of the sign of i,j,k,l and the functionB(x) takes the value 1 if the argument is zero,1 if the argument is 2(N+1),and zero otherwise. The selection rule for the couplings, which simplies theanalysis, follows from the quartic nature of the coupling (e.g., see [122]).

From (2.23) we see that frequencies spacings follow a simple sine function:they are linearly spaced (i.e. proportional to k/N) for k N and accumulatequadratically (i.e. as (k/2(N + 1))2) near the highest frequency, which liesjust below 2. For periodic boundary conditions, the frequencies are

k = 2 sin(k/N), (2.27)

which has only N/2 dierent linear frequencies, and an exact zone-boundarymode with N/2 = 2. The existence of this mode with exact alternation ofthe phase of neighboring oscillators allows one to obtain some exact solutions,which we consider in Sect. 2.4.4. For the 4 chain, the linear part of the on-sitepotential results in the m2 term in (2.24). If m2 k/N , then there is alsoquadratic accumulation of frequencies above k = m. This bunching plays asignicant role in the chaotic numerics, as described in Sect. (2.5). The FPU-Hamiltonian can also be transformed by using the harmonic normal modes toobtain the transformed Hamiltonian

H =N

k=1

12(P 2k +

2kQ

2k

)+

2N + 1

N

k,j,l=1

C(k, j, l)QjQkQl, (2.28)

which is simpler, having only a product of three summations to represent thecubic term. Furthermore, it is considerably more stable than the FPU- asit is a truncation to cubic order of an N -particle Toda lattice, as we haveconsidered in the previous subsection.

For numerical integrations if, for example, a single mode initial conditionis used, usually with all the energy in the form of potential energy, thenE = (1/2)2kQ

2k ( =

2kQ

2k/(2N)), and the oscillator equations are integrated

with their initial values given from (2.21). Due to the nonlinearity, the energydoes not remain in the initial mode but spreads through the mode spectrum,dened in terms of the instantaneous qi by the transformation in (2.2). Forenergy suciently low that there is no resonance overlap (see Sect. 2.4.1),the energy is principally conned to the initial mode falling exponentiallyto other k-values, but satisfying the selection rule as given by (2.25) andfollowing. Using perturbation theory, DeLuca et al. [55] obtained the modeenergy decay in geometric progression


Eh 2Eh2 , (2.29)

with as the initial mode and h the index of any high-frequency mode, and is the average decay ratio between modes, 2 apart, given by

=3E4

. (2.30)

The formula only holds for E 1, where resonances do not play a signicantrole. Numerical results for = 3 and 5 agreed quite well with the analyticalpredictions of (2.29) and (2.30).

The related problem of the FPU with alternating masses, as given in (2.20),has linear normal modes with frequencies

j =1 + m1 + m2 + 2m cos kj

m(2.31)

where kj = 2j/(N + 1) and 1 j N/2 (for notational convenience we usej as the mode number). The acoustic branch has 0 < j < N/2 and the opticalbranch for N/2 < j < N . The dispersion, calculated from (2.31), shows anoptical branch that moves to higher frequencies and attens as m is decreased;for example, the frequency separation of the minimum optical and maximumacoustic frequency is =

2(

(1/m) 1), while the optical frequency

spread is h =

2/m(1 + m 1) m/2, m 1. The implication for

isolated modes in the thermodynamic limit is discussed in Sect. 2.4.2.Depending on the nonlinear forces, there are implications for the stability

of nonlinear structures for the various forms of the linear modes. A weak spring( < 0) in the FPU puts the nonlinear solution in the acoustic band whichcan then radiatively couple to the linear modes, destroying nonlinear stability.Similarly, for a strong spring ( > 0) but with an optical branch, a nonlinearacoustic mode can be shifted into the optical branch where it can dissipateby loosing energy to that branch. The various treatments of these phenomenaare a major area for study (see, e.g., [84]), beyond the subject matter of thisreview. However, the phenomena will reappear in various subsequent sections.

2.3 Formulations: Methods of Numerical Analysis

Apart from the exception of integrable cases, most of the models of oscillatorchains introduced in Sect. 2.2 require numerical investigation. The choice ofsuitable observables is then crucial to point out features of mathematical andphysical interest. In this section, we introduce the description of indicatorsconcerning both dynamical and statistical properties. In general, they areinspired by generalizations or extensions of the thermodynamic concept ofentropy.


2.3.1 Measurement of Chaos Indicators

A quantitative characterization of the oscillator chains is their degree ofchaoticity. This is measured by the largest Lyapunov exponent, whose pos-itivity can be a hint to the possible equipartition of the energy among thedegrees of freedom. Let us briey recall that if

xi = X i(x1 . . . x2N ) (2.32)

is a generic dynamical system, the tangent dynamics to this ow is de-scribed by

di

dt= Jik(x(t))k , (2.33)

where Jik = X i/xk, and the largest Lyapunov exponent is given by

= limt

1tln(t)(0) (2.34)

for almost all choices of (0), under rather general assumptions. If x =(q1, . . . , qN , p1, . . . , pN), with X i = (H/pi) for i = 1, . . . , N and X i =(H/qi) for i = N + 1, . . . , 2N the dynamical system (2.32) represents aHamiltonian ow. The corresponding tangent vector is = (1, . . . , 2N ) (1q , . . . , Nq , N+1p , . . . , 2Np ), and, by setting [x(t), (t)] {T J [x(t)] +T JT[x(t)] }/ 2T = [T + T]/2T = (d/dt) ln(T)1/2 ddt ln , thiscan be formally expressed as a time average

= limt

1t

t

0

d [x(), ()] . (2.35)

Now we want to specify the more general concept of KolmogorovSinaientropy [23, 24] associated with the Lyapunov exponents and discuss its rele-vance. Besides the largest Lyapunov exponent , in a dynamical system madeof N degrees of freedom, each one described by a pair of canonical coordinates(position and momentum) one can dene a spectrum of Lyapunov exponents,i, where the index i = 1, , 2N labels the exponents from the largest to thesmallest one. An eective algorithmic procedure for evaluating the spectrumof Lyapunov exponents is discussed in [152]. Beyond rigorous mathematicaldenitions, an interpretation of the Lyapunov spectrum can be obtained byconsidering that the partial sum hn =

ni=1 i (n 2N) measures the av-

erage exponential rates of expansion, or contraction, of a generic volume ofgeometric dimension n in phase space. Accordingly, h1 = 1 is equivalentto the denition given in (2.34), since a one-dimensional volume is a generictangent segment in phase space; h2 = 1 + 2 gives the divergence rate of asurface; h2N =

2Ni=1 i is the average divergence rate of the whole phase

space. In Hamiltonian systems, according to Liouvilles theorem, any volume


in phase space is conserved and h2N = 0. Moreover, for each i > 0 there ex-ists 2Ni+1 = i.1 Chaotic evolution implies that a small region in phasespace (for instance, the volume identifying the uncertainity region around aninitial condition) is expanded and contracted with exponential rates alongdierent directions in phase space. After a time of the order 1/ the distancebetween two innitesimally close initial conditions will have the size of theaccessible phase space; accordingly, we have no means of predicting where theimage of an initial point is in phase space by knowing the image of an initiallyclose-by point, even if after a long time these points will eventually come againclose to each other (for a detailed discussion see [123]). A very important con-ceptual achievement is that the mechanical description of a chaotic evolutioncan be replaced by a description in terms of a probability distribution onphase space which is invariant under time evolution and which allows one todene a metric entropy h. The mathematical details go beyond the scope ofthis manuscript; see [123, 124, 125, 126]. For our purposes, it is importantto mention that Pesin later proved, under rather general assumptions, thatthere exist a remarkable relation between Kolmogorovs metric entropy andthe positive component of the Lyapunov spectrum [127]:

h =

j,s.t.j>0

j , (2.36)

where the sum extends over all the positive Lyapunov exponents. This for-mula can be applied to the study of the dynamics of Hamiltonian systems,like the FPU chain. In this respect, it is particularly interesting to check thisformula in the thermodynamic limit, in which the number of oscillators tendsto innity. In general, this limit does not commute with the limit t in (2.5), i.e. the measurement of and h may depend on the order in whichthese limits are performed. Numerical evidence of the existence of a limitcurve for the spectrum of Lyapunov exponents in the thermodynamic limitfor the FPU chain was later obtained ([128]; see also Fig. 2.2 ). Further numer-ical evidence of the existence of such a limit for a variety of physical systemshave been subsequently obtained. However, a rigorous mathematical proof isstill lacking, although some attempts in this direction exist [129, 130]. Thevalue of h is expected to depend on some typical parameters, like the energydensity for a Hamiltonian chain of oscillators. For instance, the Lyapunovspectrum of the FPU- model shown in Fig. 2.2 is obtained for = 10,which is suciently large to yield a strongly chaotic dynamics. By decreas-ing suciently to enter the almost-recurrent dynamical regime observed by

1 For each conserved quantity like the energy, momentum etc., there is a pair of con-jugated exponents that are zero. Stated dierently, each conservation law amountsto a geometrical constraint that limits the access of the trajectory to a subman-ifold of phase space. Integrability has the consequence that all i are zero, i.e.there can be as many conservation laws as the number of degrees of freedom; theconverse is in general not true.


0 0.5 1i/N

0

0.1

0.2

0.3

i

N = 8N = 16N = 32N = 64

Lyapunov spectrum E/N = 10

Fig. 2.2. The spectrum of positive Lyapunov exponents of the FPU- model fordierent chain lengths, from 8 up to 64 oscillators

Fermi, Pasta, and Ulam in their original numerical experiment, the shape ofthe spectrum also changes signicantly. In this weakly chaotic regime, themaximum Lyapunov exponent is found to decrease and the positive compo-nent of the Lyapunov spectrum approaches the horizontal axis. Still the onlynull exponents are those corresponding to the conserved quantities, althoughthe others take signicantly smaller values and the value of h is drasticallyreduced. According to this description, one is led to conclude that, in thethermodynamic limit, all possible chaotic degrees of freedom should remainchaotic for arbitrarily small values of , despite that beyond a certain valueit will become practically impossible to distinguish them from zero. In thisrespect, h cannot provide a characterization of the weakly chaotic regime interms of an eective number of active degrees of freedom, as discussed inthe following Section. Nonetheless, the Lyapunov analysis can provide a clearquantitative characterization of the strong and weak chaotic regimes observedin the FPU-chain. Actually, the maximum Lyapunov exponent of the FPU-model has been analytically estimated [131] on the basis of the geometricalapproach, sketched in Sect. 2.4.7. It has been found that there is a transitionvalue of the energy density, c, at which the scaling of with changes froma strong -scaling, () 2, to a weaker one () 1/4. The numerics isgiven in Sect. 2.5 and the calculation of the scaling is outlined in Sect. 2.8.This steep scaling of () below c implies that the typical relaxation time,i.e. the inverse of , may become exceedingly large for very small values of. It is worth stressing that this result seems independent on the size N of


the system, thus indicating that the dierent relaxation regimes represent astatistically relevant eect.

2.3.2 Equipartition Indicators: Information Entropy, EectiveNumber of Modes

In the numerical experiment by Fermi, Pasta and Ulam, the initial energywas placed in a single low-k mode and the authors aimed at studying howthis energy would eventually ow to the other modes. The description of thedynamics in terms of Fourier modes was a natural approach at least for smallspecic energy, despite the fact that they are not the proper modes of thechain. They expected that the nonlinearity would yield a fast decay towardsequipartition of the energy among the Fourier modes as a natural conditionto be fulllled at thermodynamic equilibrium. The existence of the two dy-namical regimes in the FPU problem for low and high values of the energydensity, , has been characterized in this context by introducing a suitableequipartition indicator among the Fourier modes [51, 52]. This indicator isinspired by information entropy, but, at variance with Kolmogorvs metricentropy, it relies upon a heuristic denition. In a chain made of N oscillatorswith periodic boundary conditions there are N/2 independent Fourier modes.A spectral entropy S(t) can be dened as

S(t) = N/2

n=1

pn(t) ln pn(t) , (2.37)

where pn(t) = En(t)/

n En(t), En(t) being the harmonic energy of theFourier mode with wave vector kn = 2n/N at time t. When only a sin-gle Fourier mode is excited S(t) vanishes, and it takes its maximum valueSmax = ln(N/2) when equipartition of the energy among the Fourier modesis obtained. Numerical studies showed that this quantity exhibits good sta-tistical properties, while it can describe the approach to energy equipartitionstarting from either single-mode or multimode initial excitations. To comparechains of dierent lengths, a normalized quantity was dened:

(t) =Smax S(t)Smax S(0) . (2.38)

Notice (t) tends towards zero when the system approaches equipartition andthat it keeps a value close to 1 when the initial spectral entropy is maintainedduring time evolution. In the long time limit (t) was found numerically toapproach an asymptotic average value , which was used for identifying theequipartition thresholds of the FPU- and - models [51, 52]. Moreover, ithas been also observed that the very dynamics of (t) provides a qualitativecharacterization of the dierent dynamical regimes observed in these chainmodels [132]. The regular, quasi-recurrent dynamics of or of ne = Ne/N


(see below) observed for small values of turns to a fast decay towards small (ne 1) for large values of .

A more physically transparent measure is what we call the eective numberof modes containing energy, which can be dened as

Ne expS, (2.39)which is conveniently normalized as

ne = Ne/N . (2.40)

For oscillators, the same denitions (2.37)(2.40) can be used, with the energyof each oscillator taken directly from the Hamiltonian, by assigning half of thedierence potential to each neighbor, to obtain the normalized eective num-ber of oscillators containing energy, nosc, which we will use in the numericsfrom short-wavelength mode initial conditions. The instantaneous values ofne do not asymptote to one, at equipartition, due to uctuations. A simpli-ed calculation of the eect of uctuations introduces a deviation ei fromequipartition ei = ei + ei. Expanding the logarithmic function in S in (2.37)as ln(1+ei/ei) = ei/ei(1/2)(ei/ei)2 and performing the summation overi yields

ne = nosc =1N

exp{Ne ln eN e2/(2e)} = exp{N e2/(2e)}. (2.41)

Taking e = 1/N and making the assumption of normal statistics, that foreach normal mode e2 = e2 (this is conrmed by calculations), we see thatN cancels giving an asymptotic value ne = nosc = exp(0.5) = 0.61, atequipartition, for both modes and oscillators. More accurate calculations havebeen made separately for modes and oscillators, including the nonlinear termsin the oscillator calculation, yielding at equipartition, ([92] Appendix D),

ne = 0.65 nosc = 0.74. (2.42)

To obtain some smoothing of the numerical values of ne(t) and nosc(t), vari-ous short-time averages of these quantities have been used, yielding somewhatdierent values from those predicted in (2.42).

2.4 Formulations: Analytic, Low-Energyand Short-Time Results

2.4.1 Transformations and Low-Dimensional Calculations

We have seen in Sect. 2.2 that a transformation to the coordinates of harmonicnormal modes decouples the modes if only linear forces are present. For smallvalues of E, the smaller nonlinear terms couple all of the modes together.


Taking the FPU- system with energy initially placed in a long-wavelengthmode, which we consider here, the selection rule for the couplings results in ageometric progression of the energy fall-o to shorter-wavelength modes [55].The strongest interactions are therefore among neighboring modes, with theinitial energy in a long-wavelength mode interchanging energy most stronglywith its nearest neighbors. The resulting beat oscillations, as observed nu-merically in the original and much subsequent work (see Fig. 2.1), involvedprimarily a few modes. The predominant localization among a few modes al-lows a useful investigation of a reduced problem, involving some minimumnumber of modes. To look at resonance overlap, a four-mode subsystem isexamined, which contains two three-mode resonances. This was done for thesine-Gordon chain by Goedde et al. [54] and then in more detail for the FPU-chain by DeLuca et al. [55]. Summarizing the analytic method, a transforma-tion of the four-mode Hamiltonian to action-angle variables exhibits two slowangles of the major resonances s = 1 + 3 22 and sp = 2 + 4 23.A second transformation is performed to the new variables s and sp fol-lowed by employing the method of averaging over the two remaining fastangles. The resultant averaged Hamiltonian has two additional approximateconstants of the motion, which are the actions related to the averaged-overangles and thus is reduced to two freedoms. The resulting Hamiltonian hasthe approximate form

H4 =( N

)2E

[ (Js + Jsp)/4Jc

+R

8J2c(3J2s + 3J

2sp 4JsJsp + JcJs 2JcJsp)

+R

16J2c

J1J2J3J4 cos(sp + s)

+R

16J2c(

J1J22J3 cos s +

J2J23J4 cos sp)]

, (2.43)

where Jc, Jd, Js, and Jsp are the transformed actions and J1, J2, J3, and J4are the original actions, related to the transformed actions by the canonicaltransformation

J1 = Js (2.44)J2 = Jsp (2.45)J3 = Js 2Jsp + Jc (2.46)J4 = Jsp + Jd . (2.47)

Jc and Jd are new constants of the motion, resulting from the averaging,and Jc was chosen such that Jc = E/ , i.e. the action corresponding tothe initial energy, primarily in mode . The concept of resonance overlap istaken from low-dimensional chaos theory, which considers separately the phasespace motion H(Js, s) with Jsp = const. and H(Jsp, sp) with Js = const.,


with overlap being the condition that for some values of s and sp we obtainJs = Jsp. The variable actions Js and Jsp are numerically studied by lookingat the phase space of one degree of freedom in a surface of section of theother freedom, with area lling trajectories indicating resonance overlap. Theoverlap is governed by

R = (N + 1)62

E 1 , (2.48)

where R measures the ratio of nonlinear to linear energy in the resonantdegrees of freedom, analogous to the energy ratios used to calculate the modeoverlap condition in (2.7). As in that calculation, from our understanding oflow-dimensional chaos, we expect signicant stochasticity to appear for R > 1.R = 1 has recently been shown to be the transition to instability for periodicsolutions of the full chain [133].

The results for four R-values are given in Fig. 2.3, showing the transitionto stochasticity in the reduced system. From the same four-mode calculation,the frequency of a typical resonant trajectory is given by

B E( N

)2(2.49)

0.16R = 0.5

R = 4.5

R = 1.5

R = 10

0.12

J s J sJ s

0.08

0.04

0

0.16

0.12

J s 0.08

0.04

0

0.16

0.12

0.08

0.04

0

0.16

0.12

0.08

0.04

0

0 0.5 1.5 2s s

ss

0 0.5 1.5 2

0 0.5 1.5 2

0 0.5 1.5 2

Fig. 2.3. Surfaces of section of the averaged H4 system in formula (2.43) with twoslow angles and six initial conditions per picture, we plot Js vs. s


with a constant of order unity, dependent on the particular initial conditions.(Here and afterward, we use the approximation N + 1 N .)

The transition to stochasticity in a reduced system is neither necessaryto ultimately reach equipartition, nor sucient to produce equipartition onnumerically observable time-scales. We note from (2.48), as the number offreedoms increases overlap occurs at decreasing energy. However, from (2.49),we see that the stochasticity also exhibits itself on increasingly slow time-scales. Furthermore, there is a competition between local resonance overlap,which spreads energy among neighboring modes, and the process of Arnolddiusion, which transports energy along guiding resonances to modes in otherparts of the phase space. This latter process is exponentially slow at lowenergy. Although a rigorous upper bound on Arnold diusion has the formgiven in (2.3), this does not determine the diusion rate from the long wave-lengths to the short wavelengths. The appropriate calculation is made froma three-resonance model ([13]; see also [15], Sects. 6.1 and 6.2). We have al-ready considered the two resonances, which produce the local stochasticity.The third resonance, called the guiding resonance, links two short-wavelengthmodes to the low frequencies via the selection rule (2.25). Again, following[55], the calculation yields a rate of energy increase in the short-wavelengthmode proportional to exp(/), with B/h where h is the short-wavelength resonance frequency. Thus we expect the diusion to be numeri-cally observable if B > h, i.e. the low frequency beat becomes comparableto a high-frequency resonance that it can couple to, that is, one for whichB = 0 in (2.25). The smallest h (largest ) is h = (/N)2. Substitutingfor B from (2.49), together with this h yields the inequality

E > 1 (2.50)

for diusion along resonances to compete with diusion across resonances.Here, as in all other equations E appears as a product, which measures thenonlinearity. The implications of (2.50) can be seen in numerical calculationsin Fig. 2.4 at small values of R for some relatively small oscillator chains, forwhich the lower edge gives a long-time asymptotic value of Ne . Consideringthat for R > 1 there is strong local coupling among modes, then as R increasesand the energy interchange spreads to more modes, there is an increase ofNe R, given by this lower edge. However, at some value of E = Ec,satisfying (2.50) the values of Ne leave this asymptote, and, in fact, approachequipartition over longer times. This scaling, rst found numerically in [56],is physically explained by the direct transfer of energy through the guidingresonances to high-frequency modes (see [55] for a more detailed calculation).We illustrate the spreading to higher modes in Fig. 2.5 at R = 2.9 for N =32, below the Ec transition as found in Fig. 2.4. The increase in energy insome high-frequency modes, specied from the selection rules, is above thebackground, but does not increase with time. We will contrast this resultwith the spectrum for E > Ec in Fig. 2.9, which approaches equipartition astime increases.


10

7.5

5Nef

f

2.5

00 2.5 5 7.5 10

R

Fig. 2.4. Eective number of modes Ne vs. R after t = 2000(3/((/N)2); pluses

N = 16; crosses N = 32; circles N = 64

As with the FPU- oscillator chain, the FPU- can also be analyzed interms of overlapping resonances to determine the onset of large stochastic lay-ers among the long-wavelength modes. Because the nonlinear term is cubic,rather than quartic, the resonances are simpler, involving only three terms,and the scaling, with , for resonance overlap, is dierent. Shepelyansky [62]has used the same averaging procedure as described in the four-mode approxi-mation of the FPU-, to analyze the FPU- chain, obtaining the Hamiltonian

0.488 +++

++

++

+ ++

++

++

+ ++

++

++ +

++ + + + +

++

++

4.882

Log

e i

9.277

13.671

18.0660 8 16 24 32

Mode

Fig. 2.5. Log of average energies at R = 2.9 for N = 32 (E = 1.4) after t =2000(N/)2


H =N

k=1

kIk +

2N + 1

N

k1,k2,k3=1

(k1k2k3Ik1Ik2Ik3)1/2

cos(k3 k2 k1)k3,k1+k2 (2.51)

where all the angles have been averaged over, except the resonant ones forwhich k3 = k1 + k2 in the long-wavelength spectrum. For these wavenumbersfrom (2.23), k k/(N + 1) such that 3 1 + 2. The Is and s arethe action-angle variables, as in (2.43), before the nal transformation to theresonant coordinates. Because of the lower cubic products, Shepelyansky wasable to examine the full Hamiltonian, and after making two simplifying furthertransformations he derived the approximate chaos border at long wavelengths, N , where is the k-value at the center of the resonance

N3/2E1/2/2 > 1. (2.52)

Comparing (2.52) with (2.48), which has R > 1 for resonance overlap, we seethat the scaling with the perturbation strengths or are the same, as isthe scaling of N2 if we substitute for the energy density = E/N inboth cases. Thus for xed energy density (xed temperature), both formulaspredict a resonant transition to local chaos in the thermodynamic limit, N . The energy-dependence with quartic or cubic nonlinearities is, of course,dierent. Shepelyansky investigated the transition of (2.52) numerically, usingthe largest Lyapunov exponent, nding reasonable agreement. He also ts thedistribution of linear mode energies to the distribution

Ek 1kc exp(k/kc c) + 1 (2.53)

(with the best t for c = 2.65) such that kc is a measure of the numberof modes containing energy, similar to Ne , but for early times for whichthe energy distribution still decreases exponentially with mode number, i.e.the energy has not signicantly diused to the high frequencies through theArnold Web. The numerical estimate for the scaling is

kc (N32E)1/4 (2.54)

which the author was able to predict analytically. This is contrasted with theresult from Fig. 2.4, which indicates that

Ne (NE)m , (2.55)

i.e. is governed by the number of modes that can satisfy the local overlapcondition R > 1 with R given in (2.48). Shepelyansky [62] has analyticallyestimated m = 1/2.

The FPU- model can be obtained as a third order truncation of the powerseries expansion of the Toda lattice potential, dened by the Hamiltonian:


H(p, q) =N

k=1

p2k2

+a

b

N

k=1

[exp(b(qk+1 qk)) + b(qk+1 qk) 1] . (2.56)

Since the Toda lattice is integrable, i.e. does not exhibit stochastic behavior,the FPU- is more stable than the FPU-. However, because the nonlinearpotential is cubic the trajectories become unbounded at high energy. There-fore, it is restricted to examining low-energy phenomena, as was describedabove. Using (2.35) for calculating for neighboring trajectories, and choos-ing the constants a and b in (2.56) to correspond to the FPU- given in (2.1),Pettini and co-workers [61] compared the variation with time of the inte-grable and nonintegrable systems, with the result as shown in Fig. 2.6. Theinitial conditions, starting on separate orbits, separate linearly (see [15]) fromwhich, calculating from (2.35) over short times, a large Lyapunov exponent isobtained. However this eect continually diminishes in the averaging processand, after a long time, only an average exponential divergence of the trajec-tories remains. In Fig. 2.6 we show the value of stabilizing at the averageexponent for the FPU- system, while it vanishes for the Toda system. As anaside remark, we point out that the stabilized value of , shown in this gure,is not necessarily the asymptotic value, but may correspond to a value in amore localized region of the phase space. Without exploring this possibilityin detail we note that the numerical values of presented in Sect. 2.5, andcompared to calculations in Sect. 2.8, have been obtained in a way that shouldbe close to the innite-time average.

Considerable eort has been directed toward the comparison of the FPU- chain with oscillator chains constructed from discretization of the KleinGordon equation, particularly the 4 chain, with the nonlinear term being

101107

106

105

104

103

102

101

1

101 102 103 104 105 106 107 108 109

t

1(t)

1

Fig. 2.6. Maximal Lyapunov exponent vs. time for the Toda lattice (open squares)and for the FPU- model (solid triangles) for N = 32 and = 0.0217


an on-site potential. However, little attention was given to the comparison onshorter time-scales, from long-wavelength initial conditions. Comparing thecoecients in (2.25) and (2.26), for small m, we see that the nonlinearity ismuch weaker for long wavelengths (small for the FPU potential than forthe 4 potential). The opposite holds for short wavelengths, where the s areabout 2 (for small m). Physically this is easily understood, as the forces be-tween neighboring oscillators are quite small for the FPU at long wavelengths:neighboring oscillators are in phase, with nearly the same amplitudes, whileat short wavelengths the nearly out of phase amplitudes amplies the forcesbetween them, as compared to the nonlinear self-force of the 4. The conse-quences for the times to achieve equipartition, starting from either low- orhigh-frequency initial conditions, will be presented in Sect. 2.8. The strongnonlinearity, coupled with the weak dispersion at short wavelengths, which isevident from either (2.23) or (2.24), leads to narrow structures in the oscil-lator space, which exhibit the short-time characteristics of breathers. Thesestructures called chaotic breathers (CBs) are introduced in Sect. 2.4.4, andinvestigated in some detail in Sects. 2.6 and 2.7.2.

2.4.2 The Thermodynamic Limit

The analysis described in Sect. 2.4.1 of considering a few modes which containmost of the energy, to understand the subsequent behavior, is appropriate fornite, relatively small, values of N . We have already se

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