Ergodicity of a Singly-Thermostated Harmonic Oscillator

TL;DR

This paper explores the ergodicity of a singly-thermostated harmonic oscillator, demonstrating that it is possible to achieve ergodic behavior with a single thermostat variable, bridging a gap in statistical mechanics.

Contribution

The authors develop new singly-thermostated oscillator models that are consistent with Gibbs' canonical distribution, simplifying previous multi-thermostat approaches.

Findings

Singly-thermostated models can be ergodic.

New models match Gibbs' distribution.

Visualization aids in model discovery.

Abstract

Although Nose's thermostated mechanics is formally consistent with Gibbs' canonical ensemble, the thermostated Nose-Hoover ( harmonic ) oscillator, with its mean kinetic temperature controlled, is far from ergodic. Much of its phase space is occupied by regular conservative tori. Oscillator ergodicity has previously been achieved by controlling two oscillator moments with two thermostat variables. Here we use computerized searches in conjunction with visualization to find singly-thermostated motion equations for the oscillator which are consistent with Gibbs' canonical distribution. These models are the simplest able to bridge the gap between Gibbs' statistical ensembles and Newtonian single-particle dynamics.