Why does Boltzmann's ergodic hypothesis work and when does it fail

TL;DR

This paper investigates the conditions under which Boltzmann's ergodic hypothesis holds or fails in Hermitian many-body models, emphasizing the role of zero frequency modes and providing physical insights through recurrence relations.

Contribution

It introduces a new sufficient condition involving zero frequency modes, tested with harmonic oscillator assemblies, explaining the validity and failure of the ergodic hypothesis.

Findings

Zero frequency modes are crucial for ergodicity.

The ergodic hypothesis fails when zero frequency modes are absent.

Recurrence relations effectively identify ergodic behavior.

Abstract

According to a recently given ergodic condition for Hermitian many-body models the thermodynamic limit and irreversibility are necessary but by themselves not sufficient. The sufficient condition turns out to be the existence of a zero frequency mode. It is measured by an infinite product of the recurrants from the recurrence relations method, which solves the Heisenberg equation of motion in Hermitian models. This condition has been tested with a variety of assemblies of nearest-neighbor coupled harmonic oscillators. The results provide a physical insight into why the ergodic hypothesis is valid and when it fails.