Thermalization in high-dimensional systems: the (weak) role of chaos

TL;DR

This paper investigates how high-dimensional systems thermalize, emphasizing that chaos influences the process but is not always essential, especially in integrable systems where observable-dependent thermalization occurs.

Contribution

It extends Khinchin's idea of typicality to non-equilibrium systems, analyzing the role of chaos and integrability in thermalization of oscillator chains.

Findings

In integrable systems, thermalization depends on the observable.

Chaotic systems tend to thermalize regardless of the observable.

Relaxation timescales in chaotic regimes can exceed observation times.

Abstract

In their seminal work, Fermi, Pasta, Ulam and Tsingou explored the connection between statistical mechanics and dynamical properties, such as chaos and ergodicity. Even today, seventy years later, the topic is not fully understood: while most results of statistical mechanics require the ergodic hypothesis to be rigorously proved, there are many indications that these predictions, both in and out of equilibrium, hold even in the absence of a rigorous form of ergodicity. Motivated by the above considerations, in this work we reconsider the point of view that the relevant ingredients for the validity of statistical mechanics are the large number of degrees of freedom and the choice of extensive observables, while the details of the dynamics do not play an essential role. This is the idea behind Khinchin's famous proof of the typicality of macroscopic observables at equilibrium. We extend this perspective to the context of non equilibrium, by investigating the thermalization properties of both harmonic (integrable) and nonharmonic (chaotic) oscillator chains initially prepared in out-of-equilibrium conditions. In integrable systems, thermalization occurs, or not, depending on the observable. In the chaotic regime, instead, thermalization is reached by any observable, although the relaxation timescale might be larger than the observation time.