Nonlinear Science

HAMILTONIANCHAOS andSTATISTICALMECHANICSThe specific examples of chaotic sys-

tems discussed in the main text—the lo-gistic map, the damped, driven pendu-lum, and the Lorenz equations—are alldissipative. It is important to recognizethat nondissipative Hamiltonian systemscan also exhibit chaos; indeed, Poincaremade his prescient statement concerningsensitive dependence on initial conditionsin the context of the few-body Hamil-tonian problems he was studying. Herewe examine briefly the many subtletiesof Hamiltonian chaos and, as an illustra-tion of its importance, discuss how it isclosely tied to long-standing problems inthe foundations of statistical mechanics.

We choose to introduce Hamiltonianchaos in one of its simplest incarnations,a two-dimensional discrete model calledthe standard map. Since this map pre-serves phase-space volume (actually areabecause there are only two dimensions)it indeed corresponds to a discrete ver-sion of a Hamiltonian system. Like thediscrete logistic map for dissipative sys-tems, this map represents an archetype forHamiltonian chaos.

The equations defining the standardmap are

is the analogue of the coordinate, andthe discrete index n plays the role of

torus, periodic in both p and q. For anyvalue of k, the map preserves the areain the (p, q) plane, since the Jacobian

The preservation of phase-space vol-ume for Hamiltonian systems has the veryimportant consequence that there can beno attractors, that is, no subregions oflower phase-space dimension to whichthe motion is confined asymptotically.Any initial point (pO, qo) will lie on someparticular orbit, and the image of allpossible initial points—that is, the unitsquare itself—is again the unit square. Incontrast, dissipative systems have phase-space volumes that shrink. For example,the logistic map (Fig. 5 in the main text)

terval (O, 1) attracted to just two points.Clearly, for k = O the standard map

time (n) as it should for free motion. Theorbits are thus just straight lines wrap-ping around the torus in the q direction.For k = 1.1 the map produces the orbits

shown in Figs. la-d. The most immedi-ately striking feature of this set of figuresis the existence of nontrivial structure onall scales. Thus, like dissipative systems,Hamiltonian chaos generates strange frac-tal sets (albeit “fat” fractals, as discussedbelow). On all scales one observes “is-lands,” analogues in this discrete case ofthe periodic orbits in the phase plane ofthe simple pendulum (Fig. 2 in the maintext). In addition, however, and again onall scales, there are swarms of dots com-ing from individual chaotic orbits that un-dergo nonperiodic motion and eventuallyfill a finite region in phase space. In thesechaotic regions the motion is “sensitivelydependent on initial conditions.”

Figure 2 shows, in the full phase space,a plot of a single chaotic orbit followedthrough 100 million iterations (again, fork = 1.1). This object differs from thestrange sets seen in dissipative systems inthat it occupies a finite fraction of the fullphase space: specifically, the orbit showntakes up 56 per cent of the unit area thatrepresents the full phase space of the map.Hence the “dimension” of the orbit is thesame as that of the full phase space, andcalculating the fractal dimension by thestandard method gives d f = 2. How-ever, the orbit differs from a conventionalarea in that it contains holes on all scales.As a consequence, the measured value ofthe area occupied by the orbit dependson the resolution with which this area ismeasured—for example, the size of theboxes in the box-counting method—andthe approach to the finite value at in-finitely fine resolution has definite scalingproperties. This set is thus appropriatelycalled a “fat fractal,” For our later dis-cussion it is important to note that theholes—representing periodic, nonchaoticmotion—also occupy a finite fraction ofthe phase space.

To develop a more intuitive feel for fatfractals, note that a very simple exam-ple can be constructed by using a slightmodification of the Cantor-set technique

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Nonlinear Science

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Fig. 1. Shown here are the discrete orbits ofthe standard map (for k = 1.1 in Eq. 1) withdifferent colors used to distinguish one orbitfrom another. increasingly magnified regionsof the phase space are shown, starting with thefull phase space (a). The white box in (a) is theregion magnified in (b), and so forth. Nontrivialstructure, including “’islands” and swarms ofdots that represent regions of chaotic, nonpe-riodic motion, are obvious on all scales. (Fig-ure courtesy of James Kadtke and David Um-berger, Los Alamos National Laboratory.)

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Nonlinear Science

described in the main text. Instead ofdeleting the middle one-third of each in-terval at every scale, one deletes the mid-

sulting set is topologically the same asthe original Cantor set, a calculation ofits dimension yields df = 1; it has thesame dimension as the full unit interval.Further, this fat Cantor set occupies a fi-nite fraction-amusingly but accidentallyalso about 56 per cent-of the unit inter-val, with the remainder occupied by the“holes” in the set.

To what extent does chaos exist in themore conventional Hamiltonian systemsdescribed by differential equations? Afull answer to this question would requirea highly technical summary of more thaneight decades of investigations by math-ematical physicists. Thus we will haveto be content with a superficial overviewthat captures, at best, the flavor of theseinvestigations.

To begin, we note that completely in-tegrable systems can never exhibit chaos,independent of the number of degrees offreedom N. In these systems all boundedmotions are quasiperiodic and occur onhypertori, with the N frequencies (pos-sibly all distinct) determined by the val-ues of the conservation laws. Thus therecannot be any aperiodic motion. Fur-ther, since all Hamiltonian systems withN = 1 are completely integrable, chaoscannot occur for one-degree-of-freedomproblems.

For N =2, non-integrable systems canexhibit chaos; however, it is not trivialto determine in which systems chaos canoccur; that: is, it is in general not obvi-ous whether a given system is integrableor not. Consider, for example, two verysimilar N = 2 nonlinear Hamiltonian sys-tems with equation of motion given by:

andd2x

Equation 2 describes the famous Henon-Heiles system, which is non-integrableand has become a classic example of asimple (astro-) physically relevant Hamil-tonian system exhibiting chaos. On theother hand, Eq. 3 can be separated intotwo independent N = 1 systems (by a

tegrable.Although there exist explicit calcula-

tional methods for testing for integrabil-ity, these are highly technical and gener-ally difficult to apply for large N. For-tunately, two theorems provide generalguidance. First, Siegel’s Theorem con-siders the space of Hamiltonians analyticin their variables: non-integrable Hamil-tonians are dense in this space, whereasintegrable Hamiltonians are not. Sec-ond, Nekhoroshev’s Theorem leads to thefact that all non-integrable systems have aphase space that contains chaotic regions.

Out observations concerning the stan-dard map immediately suggest an essen-tial question: What is the extent of thechaotic regions and can they, under somecircumstances, cover the whole phasespace? The best way to answer this ques-tion is to search for nonchaotic regions.Consider, for example, a completely inte-grable N-degree-of-freedom Hamiltoniansystem disturbed by a generic non-inte-grable perturbation. The famous KAM(for Kolmogorov, Arnold, and Moser)theorem shows that, for this case, thereare regions of finite measure in phasespace that retain the smoothness associ-ated with motion on the hypertori of theintegrable system. These regions are theanalogues of the “holes” in the standardmap. Hence, for a typical Hamiltoniansystem with N degrees of freedom, the

chaotic regions do not fill all of phasespace: a finite fraction is occupied by “in-variant KAM tori.”

At a conceptual level, then, the KAMtheorem explains the nonchaotic behav-ior and recurrences that so puzzled Fermi,Pasta, and Ulam (see “The Fermi, Pasta,and Ulam Problem: Excerpts from ‘Stud-ies of Nonlinear Problems’ “). Althoughthe FPU chain had many (64) nonlinearlycoupled degrees of freedom, it was closeenough (for the parameter ranges studied)to an integrable system that the invariantKAM tori and resulting pseudo-integrableproperties dominated the behavior overthe times of measurement.

There is yet another level of subtletyto chaos in Hamiltonian systems: namely,the structure of the phase space. For non-integrable systems, within every regularKAM region there are chaotic regions.Within these chaotic regions there are, inturn, regular regions, and so forth. Forall non-integrable systems with N > 3,an orbit can move (albeit on very longtime scales) among the various chaoticregions via a process known as “Arnolddiffusion.” Thus, in general, phase spaceis permeated by an Arnold web that linkstogether the chaotic regions on all scales.

Intuitively, these observations concern-ing Hamiltonian chaos hint strongly at aconnection to statistical mechanics. AsFig. 1 illustrates, the chaotic orbits inHamiltonian systems form very compli-cated “Cantor dusts,” which are nonperi-odic, never-repeating motions that wan-der through volumes of the phase space,apparently constrained only by conser-vation of total energy. In addition, inthese regions the sensitive dependenceimplies a rapid loss of information aboutthe initial conditions and hence an effec-tive irreversibility of the motion. Clearly,such wandering motion and effective ir-reversibility suggest a possible approachto the following fundamental question ofstatistical mechanics: How can one de-rive the irreversible, ergodic, thermal-

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Nonlinear Science

equilibrium motion assumed in statisticalmechanics from a reversible, Hamiltonianmicroscopic dynamics?

Historically, the fundamental assump-tion that has linked dynamics and statis-tical mechanics is the ergodic hypothesis,which asserts that time averages over ac-tual dynamical motions are equal to en-semble averages over many different butequivalent systems. Loosely speaking,this hypothesis assumes that all regionsof phase space allowed by energy con-servation are equally accessed by almostall dynamical motions.

What evidence do we have that the er-godic hypothesis actually holds for re-alistic Hamiltonian systems? For sys-tems with finite degrees of freedom, theKAM theorem shows that, in additionto chaotic regions of phase space, thereare nonchaotic regions of finite measure.These invariant tori imply that ergodicitydoes not hold for most finite-dimensionalHamiltonian sytems. Importantly, thefew Hamiltonian systems for which theKAM theorem does not apply, and forwhich one can prove ergodicity and theapproach to thermal equilibrium, involve“hard spheres” and consequently containnon-analytic interactions that are not re-alistic from a physicist’s perspective.

For many years, most researchers be-lieved that these subtleties become irrele-vant in the thermodynamic limit, that is,the limit in which the number of degreesof freedom (N) and the energy (E) go toinfinity in such a way that E/N remainsa nonzero constant. For instance, theKAM regions of invariant tori may ap-proach zero measure in this limit. How-ever, recent evidence suggests that non-trivial counterexamples to this belief mayexist. Given the increasing sophisticationof our analytic understanding of Hamilto-nian chaos and the growing ability to sim-ulate systems with large N numerically,the time seems ripe for quantitative inves-tigations that can establish (or disprove!)this belief. (For additional discussion of

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A “FAT” FRACTAL

Fig. 2. A singles chaotic orbit of the standardmap for k = 1.1. The picture was made by di-viding the energy surface Into a 512 by 512 gridand iterating the initial condition 108 times.The squares visited by this orbit are shownin black. Gaps in the phase space representportions of the energy surface unavailable tothe chaotic orbit because of various quasiperi-odic orbits confined to tori, as seen In Fig. 1.(Figure courtesy of J. Doyne Farmer and DavidUmberger, Los Alamos National Laboratory.)

this topic, see “The Ergodic Hypothesis:A Complicated Problem of Mathematicsand Physics.”)

Among the specific issues that shouldbe addressed in a variety of physicallyrealistic models are the following.

● How does the measure of phase spaceoccupied by KAM tori depend on N ?Is there a class of models with realisticinteractions for which this measure goesto O? Are there non-integrable modelsfor which a finite measure is retained bythe KAM regions? If so, what are thecharacteristics that cause this behavior?● How does the rate of Arnold diffusiondepend on N in a broad class of mod-els? What is the structure of importantfeatures—such as the Arnold web-in thephase space as N approaches infinity?● If there is an approach to equilibrium,how does the time-scale for this approachdepend on N? Is it less than the age ofthe universe?● Is ergodicity necessary (or merely suf-ficient) for most of the features we as-sociate with statistical mechanics? Can aless stringent requirement, consistent withthe behaviour observed in analytic Hamil-tonian systems, be formulated?

Clearly, these are some of the most chal-lenging, and profound, questions current-ly confronting nonlinear scientists. ■

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