Ergodic Time-Reversible Chaos for Gibbs' Canonical Oscillator

TL;DR

This paper investigates the ergodic properties of time-reversible thermostats in Gibbs' canonical oscillator models, analyzing existing models and proposing insights into achieving ergodicity with single thermostats.

Contribution

It provides a detailed analysis of ergodicity in single- and doubly-thermostated oscillator models, highlighting challenges and potential routes toward ergodic behavior.

Findings

Doubly-thermostated models can be ergodic or not, depending on parameters.

Single-thermostated models generally fail to produce ergodicity.

Phase-space analysis and Lyapunov diagnostics reveal the ergodic nature of these models.

Abstract

Nos\'e's pioneering 1984 work inspired a variety of time-reversible deterministic thermostats. Though several groups have developed successful doubly-thermostated models, single-thermostat models have failed to generate Gibbs' canonical distribution for the one-dimensional harmonic oscillator. Sergi and Ferrario's 2001 doubly-thermostated model, claimed to be ergodic, has a singly-thermostated version. Though neither of these models is ergodic this work has suggested a successful route toward singly-thermostated ergodicity. We illustrate both ergodicity and its lack for these models using phase-space cross sections and Lyapunov instability as diagnostic tools.