Thermalization in classical systems with discrete phase space

TL;DR

This paper investigates how classical systems with discrete phase spaces reach thermal equilibrium, showing that local ergodicity and pseudorandom dynamics are sufficient, and drawing parallels with quantum thermalization theories.

Contribution

It introduces a framework demonstrating that classical thermalization can occur without global ergodicity, emphasizing local ergodic behavior and spectral analysis.

Findings

Thermalization arises from effective local ergodicity, not global ergodicity.

Spectral analysis of the evolution operator reveals pseudorandom local dynamics.

A classical analog of the Eigenstate Thermalization Hypothesis is proposed.

Abstract

We study the emergence of statistical mechanics in isolated classical systems with local interactions and discrete phase spaces. We establish that thermalization in such systems does not require global ergodicity; instead, it arises from effective local ergodicity, where dynamics in a subsystem may appear pseudorandom. To corroborate that, we analyze the spectrum of the unitary evolution operator and propose an ansatz to describe statistical properties of local observables expanded in the eigenfunction basis - the classical counterpart of the Eigenstate Thermalization Hypothesis. Our framework provides a unified perspective on thermalization in classical and quantum systems with discrete spectra.