Time evolution of correlation functions and thermalization

TL;DR

This paper studies the time evolution and thermalization behavior of classical chains of anharmonic oscillators, revealing incomplete thermalization and mesoscopic dynamics, challenging traditional views on thermalization in isolated systems.

Contribution

It introduces an exact evolution equation for correlation functions and analyzes its solutions in a 1/N expansion, highlighting nonthermal stationary states in large systems.

Findings

System approaches stationary states near thermal equilibrium but retains memory of initial conditions.

Incomplete thermalization persists regardless of system size.

Challenges Boltzmann's conjecture on thermalization of macroscopic systems.

Abstract

We investigate the time evolution of a classical ensemble of isolated periodic chains of O(N)-symmetric anharmonic oscillators. Our method is based on an exact evolution equation for the time dependence of correlation functions. We discuss its solutions in an approximation which retains all contributions in next-to-leading order in a 1/N expansion and preserves time reflection symmetry. We observe effective irreversibility and approximate thermalization. At large time the system approaches stationary solutions in the vicinity of, but not identical to, thermal equilibrium. The ensemble therefore retains some memory of the initial condition beyond the conserved total energy. Such a behavior with incomplete thermalization is referred to as "mesoscopic dynamics". It is expected for systems in a small volume. Surprisingly, we find that the nonthermal asymptotic stationary solutions do not change for large volume. This raises questions on Boltzmann's conjecture that macroscopic isolated systems thermalize.