The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior

Abstract

A numerical and analytical study of the relaxation to equilibrium of both the Fermi-Pasta-Ulam (FPU) alpha-model and the integrable Toda model, when the fundamental mode is initially excited, is reported. We show that the dynamics of both systems is almost identical on the short term, when the energies of the initially unexcited modes grow in geometric progression with time, through a secular avalanche process. At the end of this first stage of the dynamics the time-averaged modal energy spectrum of the Toda system stabilizes to its final profile, well described, at low energy, by the spectrum of a q-breather. The Toda equilibrium state is clearly shown to describe well the long-living quasi-state of the FPU system. On the long term, the modal energy spectrum of the FPU system slowly detaches from the Toda one by a diffusive-like rising of the tail modes, and eventually reaches the equilibrium flat shape. We find a simple law describing the growth of tail modes, which enables us to estimate the time-scale to equipartition of the FPU system, even when, at small energies, it becomes unobservable.

31 pages, 11 figures, submitten in Chaos

References in corpus (3)

• Energy localization on q-tori, long term stability and the interpretation of FPU recurrences
• The Fermi-Pasta-Ulam problem: periodic orbits, normal forms and resonance overlap criteria
• Tail resonances of FPU q-breathers and their impact on the pathway to equipartition

Cited by in corpus (22)

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• Double scaling in the relaxation time in the $\beta$-FPUT model
• Thermalization of dilute impurities in one dimensional spin chains
• Equilibration of Quasi-Integrable Systems
• Arbitrarily Weak Nonlinearity Can Destroy the Anderson Localization
• Wave Turbulence and thermalization in one-dimensional chains
• Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU
• Universal law of thermalization for one-dimensional perturbed Toda lattices
• Behavior and Breakdown of Higher-Order Fermi-Pasta-Ulam-Tsingou Recurrences
• Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties
• Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit
• Nonlinear Topological Edge States: from Dynamic Delocalization to Thermalization
• Chopping time of the FPU $\alpha$-model
• Periodic Orbits in Fermi-Pasta-Ulam-Tsingou Systems
• Dynamical chaos in the integrable Toda chain induced by time discretization
• Energy-recurrence Breakdown and Chaos in Disordered Fermi-Pasta-Ulam-Tsingou Lattices
• Energy cascade and Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain
• Extensive packet excitations in FPU and Toda lattices
• Recurrence in a periodically driven and weakly damped Fermi-Pasta-Ulam-Tsingou chain
• Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral
• Thermalization in asymmetric harmonic chains