Heat Conduction and Entropy Production in a One-Dimensional Hard-Particle Gas

Grassberger, Peter; Nadler, Walter; Yang, Lei

doi:10.1103/physrevlett.89.180601

Cited by 172 publications

(244 citation statements)

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“…(1) is satisfied with α = 1/3. This has been confirmed by simulations of hard particle gases [16,17,18], although very large systems are required [19] and the issue is not completely settled [20]. On the other hand, numerical simulations of oscillator chains, including FPU chains, give various exponents [6,10,11,12,13] for different systems, often slightly higher than 1/3.…”

Section: Pacs Numbersmentioning

confidence: 67%

Equilibration and Universal Heat Conduction in Fermi-Pasta-Ulam Chains

2007

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It is shown numerically that for Fermi Pasta Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is shown to diverge with chain length N as N 1/3 , relevant for the much debated question of the universality of one dimensional heat conduction. The reason why earlier simulations have obtained systematically higher exponents is explained. PACS numbers:It has long been established [1] that Fermi Pasta Ulam (FPU) chains (one dimensional chains of particles with anharmonic forces between them) may not be able to achieve thermal equilibrium with seemingly reasonable initial conditions. This has interesting implications for the ergodic hypothesis, at the foundation of statistical mechanics. The results of Ref.[1] led to the discovery of solitons in continuum versions [2], and eventually an understanding of chaotic dynamics. With regard to the initial results, a vast body of work has established [3,4] that quasi-periodic solutions exist below an energy threshold, above which the system does equilibrate.Although all this work has been with closed boundary conditions, the study of FPU chains has expanded to include heat bath boundary conditions: with slightly different temperatures imposed at the two ends, the thermal conductivity can be measured and shows anomalous properties [5,6]. Since the baths at the ends are at different temperatures, the concept of equilibrium has to be extended: a local temperature (LT) that varies smoothly along the chain has to be defined. Surprisingly, this has not been fully investigated, even though the discussion of heat conductivity is in terms of Fourier's law [5] which is meaningless if the local temperature is ill-behaved. When the heat baths at the two ends have equal temperatures, one can prove analytically that the only possible steady state is the one in thermal equilibrium [7].In this paper, we demonstrate for the first time that, for FPU chains with heat bath boundary conditions, the LT behaves paradoxically on small scales. In particular, we show through numerical simulations that for FPU-β chains with alternating light and heavy masses, connected to heat baths at temperatures T L and T R at the left and right end respectively (with ∆T = T L − T R sufficiently small that the system is in the linear response regime), the kinetic temperature of the particles oscillates as one moves along the chain. The ratio of amplitude of these oscillations to ∆T /N is constant as ∆T is reduced, and increases with the chain length N : in the vicinity of the 2i'th particle, the temperature difference between heavy and light particles scales approximately as δT ∼ [∆T / √ N ]f (i/N ). Thus if one were to coarse grain over a region of the order of the mean free path, the intracell temperature uncertainty O(∆T...

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Section: Pacs Numbersmentioning

confidence: 67%

Equilibration and Universal Heat Conduction in Fermi-Pasta-Ulam Chains

2007

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“…On the other hand, it is easy to convince oneself that the maximum Lyapunov exponent is still exactly equal to zero. The argument is pretty straightforward [43]: since the collision rule is linear, Fig. 14.…”

Section: A Hamiltonian Model: Diatomic Hard-point Chainmentioning

confidence: 99%

Nonlinear Dynamics and Chaos: Advances and Perspectives

Thiel¹,

Kurths²,

Romano³

et al. 2010

Understanding Complex Systems

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Abstract. Stable chaos is a generalization of the chaotic behaviour exhibited by cellular automata to continuous-variable systems and it owes its name to an underlying irregular and yet linearly stable dynamics. In this review we discuss analogies and differences with the usual deterministic chaos and introduce several tools for its characterization. Some examples of transitions from ordered behavior to stable chaos are also analyzed to further clarify the underlying dynamical properties. Finally, two models are specifically discussed: the diatomic hard-point gas chain and a network of globally coupled neurons.

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“…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.…”

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confidence: 99%

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

2009

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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...

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Heat Conduction and Entropy Production in a One-Dimensional Hard-Particle Gas

Cited by 172 publications

References 28 publications

Equilibration and Universal Heat Conduction in Fermi-Pasta-Ulam Chains

Equilibration and Universal Heat Conduction in Fermi-Pasta-Ulam Chains

Nonlinear Dynamics and Chaos: Advances and Perspectives

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

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