Ergodic observables in non-ergodic systems: the example of the harmonic chain

TL;DR

This paper demonstrates that a non-ergodic harmonic chain exhibits ergodic behavior in the large system limit, with Maxwell-Boltzmann statistics accurately describing single particle velocities, supported by analytical and numerical evidence.

Contribution

It provides a pedagogical analysis showing ergodic-like properties in a non-chaotic harmonic chain with many degrees of freedom.

Findings

Maxwell-Boltzmann distribution describes single particle velocities

Ergodic behavior emerges in the large system limit

Relaxation time scales are characterized

Abstract

In the framework of statistical mechanics the properties of macroscopic systems are deduced starting from the laws of their microscopic dynamics. One of the key assumptions in this procedure is the ergodic property, namely the equivalence between time averages and ensemble averages. This property can be proved only for a limited number of systems; however, as proved by Khinchin [1], weak forms of it hold even in systems that are not ergodic at the microscopic scale, provided that extensive observables are considered. Here we show in a pedagogical way the validity of the ergodic hypothesis, at a practical level, in the paradigmatic case of a chain of harmonic oscillators. By using analytical results and numerical computations, we provide evidence that this non-chaotic integrable system shows ergodic behavior in the limit of many degrees of freedom. In particular, the Maxwell-Boltzmann distribution turns out to fairly describe the statistics of the single particle velocity. A study of the typical time-scales for relaxation is also provided.