Lyapunov Instability and Finite Size Effects in a System with Long-Range Forces

Latora, Vito; Rapisarda, Andrea; Ruffo, Stefano

doi:10.1103/physrevlett.80.692

Cited by 167 publications

(233 citation statements)

References 28 publications

(47 reference statements)

Supporting

Mentioning

209

Contrasting

Order By: Relevance

“…Then, the tangent dynamics of the flow, i. e. the solution of the variational equations (7), is described by the tangent map T S = ∂S/∂ x of S (some particular implementations of this approach for different physical problems can be found in [36,37]). …”

Section: Summary and Discussionmentioning

confidence: 99%

Numerical integration of variational equations

Skokos

Gerlach

2010

Phys. Rev. E

View full text Add to dashboard Cite

We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the 'tangent map (TM) method', a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton's equations of motion by the repeated action of a symplectic map S, while the corresponding tangent map T S, is used for the integration of the variational equations. A simple and systematic technique to construct T S is also presented.

show abstract

Section: Summary and Discussionmentioning

confidence: 99%

Numerical integration of variational equations

Skokos

Gerlach

2010

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…During the last years, after an earlier attempt in Ref. [29] where the classical XY model in two dimensions was considered, and its largest Lyapunov exponent was found to display some indication of the transition temperature of the Kosterlitz-Thouless phase transition, there has been a renewed interest in the study of the behaviour of Lyapunov exponents in systems undergoing phase transitions, and a number of papers have appeared: see [27,28,30] and [25,47,48,49,50,51,52,53,54,55].…”

Section: Hamiltonian Dynamics Phase Transitions and Topologymentioning

confidence: 99%

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Pettini

Casetti²,

Cerruti-Sola

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. A first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: i) a Stochasticity Threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. ii) a Strong Stochasticity Threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weakly and strongly chaotic regimes. It is stable with N . A second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. The starting of this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N , or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, a third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions. "...Fermi expressed often the belief that future fundamental theories in physics may involve non-linear operators and equations, and that it would be useful to attempt practice in the mathematics needed for the understanding of nonlinear systems. The plan was then to start with the possibly simplest such physical model and to study the results of the calculation of its long-time behavior.... The motivation then was to observe the rates of mixing and thermalization with the hope that the computational results would provide hints for a future theory. One could venture a guess that one motive in the selection of problems could be traced to Fermi's early interest in the ergodic theory..."Actually, Fermi's early interest in ergodic theory is witnessed by his contribution to a theorem due to Poincaré and thenceforth known as the Poincaré-Fermi theorem. This asserts that neither analytic (Poincaré) nor smooth (Fermi) integrals of motion besides the energy can survive a generic perturbation of an integrable system with three or more degrees of freedom, thus, in the absence of other isolating integrals of motion, any constant energy surface of these generic systems is expected to be everywhere accessible to the phase space trajectory. At this level, no hindrance to ergodicity seems to be possible. Whence the surprise of Fermi, Pasta and Ulam (FPU) when no apparent...

show abstract

“…It states that the Kolmogorov-Sinai entropy h KS is equal to the Lyapunov exponent λ for a closed ergodic 1d systems (to the sum of positive Lyapunov exponents for d > 1). At the same time, it is well known that many systems such as Hamiltonian models with mixed phase space [2], systems with long range forces [3], certain billiards [4] and one-dimensional hard-particle gas [5] have a Lyapunov exponent equal zero. While for complex systems it may be extremely difficult to determine whether the Lyapunov exponent is zero or small, due to numerical inaccuracies, it turns out that most fundamental text book examples of chaos theory may have a zero Lyapunov exponent.…”

mentioning

confidence: 99%

Pesin-Type Identity for Intermittent Dynamics with a Zero Lyaponov Exponent

2009

View full text Add to dashboard Cite

Pesin's identity provides a profound connection between entropy hKS (statistical mechanics) and the Lyapunov exponent λ (chaos theory). It is well known that many systems exhibit sub-exponential separation of nearby trajectories and then λ = 0. In many cases such systems are non-ergodic and do not obey usual statistical mechanics. Here we investigate the non-ergodic phase of the PomeauManneville map where separation of nearby trajectories follows δxt = δx0e λαt α with 0 < α < 1. The limit distribution of λα is the inverse Lévy function. The average λα is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.PACS numbers: 05.90.+m, 05.45.Ac, 74.40.+k Chaotic systems are characterized by exponential separation of nearby trajectories, which is quantified by a positive Lyapunov exponent λ [1]. Such a behavior leads to the need for statistical approaches since chaos implies our inability to predict the long time limit of a system in a deterministic fashion. A profound relation between chaos and statistical mechanics is given by Pesin's identity [1]. It states that the Kolmogorov-Sinai entropy h KS is equal to the Lyapunov exponent λ for a closed ergodic 1d systems (to the sum of positive Lyapunov exponents for d > 1). At the same time, it is well known that many systems such as Hamiltonian models with mixed phase space [2], systems with long range forces [3], certain billiards [4] and one-dimensional hard-particle gas [5] have a Lyapunov exponent equal zero. While for complex systems it may be extremely difficult to determine whether the Lyapunov exponent is zero or small, due to numerical inaccuracies, it turns out that most fundamental text book examples of chaos theory may have a zero Lyapunov exponent. Prominent examples for such weakly chaotic systems are the logistic map at the edge of chaos (Feigenbaum's point) [6] and the Pomeau-Manneville map which is used to model intermittency (originally in turbulence) [7]. If the Lyapunov exponent is zero, i.e. separation of trajectories is sub-exponential, we have a strong indication that the usual Boltzmann-Gibbs statistical mechanics is not valid. Indeed it was found that certain systems with zero Lyapunov exponents break ergodicity [8,9]. Classical entropy theory is also not applicable in this case [6,7], particularly the entropy and average algorithmic complexity grow non linearly in time [7], while for a system with a positive Lyapunov exponent they increase linearly in time. Still the situation is not hopeless from the point of view of statistical mechanics and one may consider distributions of time average observables [8,9,10,11,12].Connection between possible generalizations of usual statistical mechanics and weakly chaotic systems have attracted much attention recently. In particular, a generalized Pesin's identity for the logistic map at the edge of chaos was investigated using Tsallis statistics [6,13]. A critical discussion of this appr...

show abstract

Lyapunov Instability and Finite Size Effects in a System with Long-Range Forces

Cited by 167 publications

References 28 publications

Numerical integration of variational equations

Numerical integration of variational equations

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Pesin-Type Identity for Intermittent Dynamics with a Zero Lyaponov Exponent

Contact Info

Product

Resources

About