The Fermi-Pasta-Ulam problem and its underlying integrable dynamics: an approach through Lyapunov Exponents

Abstract

FPU models, in dimension one, are perturbations either of the linear model or of the Toda model; perturbations of the linear model include the usual $\beta$-model, perturbations of Toda include the usual $\alpha +\beta$ model. In this paper we explore and compare two families, or hierarchies, of FPU models, closer and closer to either the linear or the Toda model, by computing numerically, for each model, the maximal Lyapunov exponent $\chi$. We study the asymptotics of $\chi$ for large $N$ (the number of particles) and small $\epsilon$ (the specific energy $E/N$), and find, for all models, asymptotic power laws $\chi \simeq C{\epsilon }^a$, $C$ and $a$ depending on the model. The asymptotics turns out to be, in general, rather slow, and producing accurate results requires a great computational effort. We also revisit and extend the analytic computation of $\chi$ introduced by Casetti, Livi and Pettini, originally formulated for the $\beta$-model. With great evidence the theory extends successfully to all models of the linear hierarchy, but not to models close to Toda.

Cited by in corpus (14)

• Equilibration of Quasi-Integrable Systems
• Universal law of thermalization for one-dimensional perturbed Toda lattices
• Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit
• Thermalization of local observables in the $\alpha$-FPUT chain
• Korteweg-de Vries and Fermi-Pasta-Ulam-Tsingou: asymptotic integrability of quasi unidirectional waves
• Chaotic dynamics in a quantum Fermi-Pasta-Ulam problem
• Effect of pressure on thermalization of one-dimensional nonlinear chains
• Fluctuation Theorem for Quasi-Integrable Systems
• Dynamical chaos in the integrable Toda chain induced by time discretization
• On the role of the Integrable Toda model in one-dimensional molecular dynamics
• Adiabatic Gauge Potential as a Tool for Detecting Chaos in Classical Systems
• Thermalization in asymmetric harmonic chains
• From Near-Integrable to Far-from-Integrable: A Unified Picture of Thermalization and Heat Transport
• Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems