Thermalization of local observables in the $\alpha$-FPUT chain

TL;DR

This paper investigates how local variables in the $eta$-FPUT chain equilibrate over time, highlighting differences from normal mode equipartition and linking thermalization times to chaos and initial condition distributions.

Contribution

It introduces a detailed analysis of local variable thermalization, exploring effects of initial condition width, chaos, and integrability on equilibration times.

Findings

Thermalization times vary significantly with initial condition width.

Chaos, measured by Lyapunov exponents, correlates with thermalization speed.

Ensemble averaging can induce thermalization in the integrable Toda chain.

Abstract

Most studies on the problem of equilibration of the Fermi-Pasta-Ulam-Tsingou (FPUT) system have focused on equipartition of energy being attained amongst the normal modes of the corresponding harmonic system. In the present work, we instead discuss the equilibration problem in terms of local variables, and consider initial conditions corresponding to spatially localized energy. We estimate the time-scales for equipartition of space localized degrees of freedom and find significant differences with the times scales observed for normal modes. Measuring thermalization in classical systems necessarily requires some averaging, and this could involve one over initial conditions or over time or spatial averaging. Here we consider averaging over initial conditions chosen from a narrow distribution in phase space. We examine in detail the effect of the width of the initial phase space distribution, and of integrability and chaos, on the time scales for thermalization. We show how thermalization properties of the system, quantified by its equilibration time, defined in this work, can be related to chaos, given by the maximal Lyapunov exponent. Somewhat surprisingly we also find that the ensemble averaging can lead to thermalization of the integrable Toda chain, though on much longer time scales.