The Fermi–Pasta–Ulam Problem and Its Underlying Integrable Dynamics: An Approach Through Lyapunov Exponents

Benettin, Giancarlo; Pasquali, Stefano; Ponno, Antonio

doi:10.1007/s10955-018-2017-x

Cited by 29 publications

(43 citation statements)

References 45 publications

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“…The FPUT-α model is a good example, for which supporting evidence from various aspects, e.g. by a normal mode approach [50],and by the Lyapunov exponent analysis [51][52][53], has been found. In particular, in the study of thermalization problem [35],it was found that the dynamics of the FPUT is almost indistinguishable from the Toda dynamics in the initial stage.…”

Section: Introductionmentioning

confidence: 63%

Universal law of thermalization for one-dimensional perturbed Toda lattices

Zhang

Zhao

2019

New J. Phys.

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The Toda lattice is a nonlinear but integrable system. Here we study the thermalization problem in one-dimensional, perturbed Toda lattices in the thermodynamic limit. We show that the thermalization time, T eq , follows a universal law; i.e. T eq ∼ò −2 , where the perturbation strength, ò, characterizes the nonlinear perturbations added to the Toda potential. This universal law applies generally to weak nonlinear lattices due to their equivalence to perturbed Toda systems.

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Section: Introductionmentioning

confidence: 63%

Universal law of thermalization for one-dimensional perturbed Toda lattices

Zhang

Zhao

2019

New J. Phys.

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“…[6] we also integrate the initial ensemble with the integrable Toda potential, Eq. (7). Here, the system reaches a terminal, non-ergodized state.…”

Section: Quasi-static Statesmentioning

confidence: 99%

“…However, in such systems the corresponding Lyapunov time is usually much shorter than the characteristic time of diffusion within the space of quasi-conserved quantities. For example, a FPU chain of size N = 1024 and energy density ǫ = E/N = 10 −4 has a Lyapunov time of 10 6 , much shorter than the relaxation time of 10 10 [6,7]. Another example is the Solar System, there, the Lyapunov time is 5-10 Myrs, whereas the stability time is over 5 Gyrs [8].…”

Section: Introductionmentioning

confidence: 99%

Equilibration of quasi-integrable systems

Goldfriend

Kurchan

2019

Phys. Rev. E

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We study the slow relaxation of isolated quasi-integrable systems, focusing on the classical problem of Fermi-Pasta-Ulam-Tsingou (FPU) chain. It is well-known that the initial energy sharing between different linear-modes can be inferred by the integrable Toda chain. Using numerical simulations, we show explicitly how the relaxation of the FPU chain toward equilibration is determined by a slow drift within the space of Toda's integrals of motion. We analyze the whole spectrum of Toda-modes and show how they dictate, via a Generalized Gibbs Ensemble (GGE), the quasi-static states along the FPU evolution. This picture is employed to devise a fast numerical integration, which can be generalized to other quasi-integrable models. In addition, the GGE description leads to a fluctuation theorem, describing the large deviations as the system flows in the entropy landscape. *

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“…Its simplest realization in atomic chains probed as a modeling system for irreversible dynamics was considered in the celebrated work by Fermi, Pasta and Ulam [ 25 ] (FPU), where the quasi-periodic behavior has been discovered for the evolution of the initial excitation instead of irreversible energy equipartition. Despite over sixty years of investigations of the FPU problem, its complete understanding remains a challenge [ 26 , 27 , 28 , 29 ].…”

Section: Introductionmentioning

confidence: 99%

“…The consideration is restricted to quantum mechanical systems. It has been suggested that the threshold energy separating localized and chaotic states decreases with the system size [ 27 , 28 , 29 , 37 , 38 , 39 , 40 ]. This leads to the reduction of thermal energy below the vibrational quantization energy, which makes quantum effects inevitably significant for sufficiently large molecules.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Dynamics in a Quantum Fermi–Pasta–Ulam Problem

Burin

Maksymov

Schmidt

et al. 2019

Entropy

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We investigate the emergence of chaotic dynamics in a quantum Fermi -Pasta -Ulam problem for anharmonic vibrations in atomic chains applying semi-quantitative analysis of resonant interactions complemented by exact diagonalization numerical studies. The crossover energy separating chaotic high energy phase and localized (integrable) low energy phase is estimated. It decreases inversely proportionally to the number of atoms until approaching the quantum regime where this dependence saturates. The chaotic behavior appears at lower energies in systems with free or fixed ends boundary conditions compared to periodic systems. The applications of the theory to realistic molecules are discussed.determines the localization threshold. The threshold energy can be redefined in terms of the critical temperature corresponding to that energy.The knowledge of the localization threshold for an individual molecule is significant since the vibrational relaxation changes dramatically depending on whether the energy of the molecule is lower or higher than the threshold [5,10,32,33]. In the latter case the vibrational relaxation follows standard Fermi Golden rule kinetics [34], while in the localized regime it is much slower. Therefore, the present work is focused on the localization threshold and its dependence on system size (number of atoms) and the strength of anharmonic interaction.Since the properties of a molecule can be sensitive to its shape the consideration is restricted to the simple linear chain of atoms coupled by anharmonic interactions identical to the FPU problem [25]. This problem is relevant for the energy relaxation and transport in polymer chains used in the modern heat conducting devices [3,32,35,36]. The anomalous increase of a thermal conductivity there with the system size suggests a very slow thermalization or even the lack of one [36]. The results for the FPU problem can be qualitatively relevant for the analysis of more complicated molecules.The consideration is restricted to quantum mechanical systems. It has been suggested that the threshold energy separating localized and chaotic states decreases with the system size [27][28][29][37][38][39][40]. This leads to the reduction of thermal energy below the vibrational quantization energy, which makes quantum effects inevitably significant for sufficiently large molecules.The paper is organized as follows. The FPU problems with different boundary conditions are formulated and briefly discussed in Sec. 1. The analysis of localization is performed combining analytical (Sec. 2) and numerical (Sec. 3) approaches for the FPU problems with different boundary conditions. Both approaches are reasonably consistent with each other and led to the predictions of analytical dependencies of localization threshold on system parameters that are discussed in Sec. 4 for organic molecules. The methods and brief conclusions are formulated in Secs. 5, 6.

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The Fermi–Pasta–Ulam Problem and Its Underlying Integrable Dynamics: An Approach Through Lyapunov Exponents

Cited by 29 publications

References 45 publications

Universal law of thermalization for one-dimensional perturbed Toda lattices

Universal law of thermalization for one-dimensional perturbed Toda lattices

Equilibration of quasi-integrable systems

Chaotic Dynamics in a Quantum Fermi–Pasta–Ulam Problem

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