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

TL;DR

This paper investigates the relaxation dynamics of the Fermi-Pasta-Ulam and Toda models, revealing a two-stage process from initial geometric growth to eventual diffusive equipartition, with implications for understanding thermalization timescales.

Contribution

It provides a combined numerical and analytical analysis of the FPU and Toda models, highlighting the two-stage relaxation process and estimating the timescale for energy equipartition.

Findings

Initial short-term dynamics are similar for FPU and Toda models.

The Toda system's spectrum stabilizes to a q-breather profile.

FPU system eventually reaches equilibrium through diffusive tail growth.

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.