Equilibration of Quasi-Integrable Systems

TL;DR

This paper investigates the slow relaxation process of quasi-integrable systems, specifically the FPU chain, by analyzing Toda integrals of motion and employing GGE, revealing new insights into their equilibration dynamics.

Contribution

It introduces a novel numerical method based on Toda integrals and GGE to understand and simulate the slow relaxation in quasi-integrable systems.

Findings

Relaxation is governed by a slow drift in Toda's integrals of motion.

GGE accurately describes quasi-static states during FPU evolution.

A new fast numerical integration method is developed for these systems.

Abstract

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.