Thermalization in classical systems with discrete phase space

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.

Figures (3)

• Figure 1: Scaling of mean distance $d_{\Lambda}$ (rescaled by $\sqrt{q^{|\Lambda|}-1}$) with the mean orbit length $T$ in Model I, shown for various subsystem sizes. The largest $T$ corresponds to $L=28$. Inset: cumulative probability density function of fluctuations in the configuration frequencies, Eq. \ref{['chi']}, compared to a Gaussian fit.
• Figure 2: The finite-size $G$-function, cf. Eq.\ref{['G-function']}, for Model I for different system sizes $L$, shown for a subsystem of size $|\Lambda|=2$ and frequency window $\delta \omega = 0.05$.
• Figure 3: Mean distance $d_{\Lambda}$ as a function of the system size $L$ in Model II, shown for a subsystem of size $|\Lambda| =2$.