Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator and the 2014 Ian Snook Prize

TL;DR

This paper examines the ergodicity of thermostated harmonic oscillators, highlighting the challenges and complexities in ensuring these systems accurately reproduce canonical distributions, and proposes methods to test ergodicity more effectively.

Contribution

The paper reviews existing thermostated oscillator models, discusses their ergodic properties, and introduces new testing methods to evaluate ergodicity more convincingly.

Findings

Thermostated harmonic oscillators often exhibit non-ergodic behavior.

Existing approaches show complex phase space structures like islands and chaos.

Proposes new methods for testing ergodicity in small systems.

Abstract

Shuichi Nos\'e opened up a new world of atomistic simulation in 1984. He formulated a Hamiltonian tailored to generate Gibbs' canonical distribution dynamically. This clever idea bridged the gap between microcanonical molecular dynamics and canonical statistical mechanics. Until then the canonical distribution was explored with Monte Carlo sampling. Nos\'e's dynamical Hamiltonian bridge requires the "ergodic" support of a space-filling structure in order to reproduce the entire distribution. For sufficiently small systems, such as the harmonic oscillator, Nos\'e's dynamical approach failed to agree with Gibbs' sampling and instead showed a complex structure, partitioned into a chaotic sea, islands, and chains of islands, that is familiar textbook fare from investigations of Hamiltonian chaos. In trying to enhance small-system ergodicity several more complicated "thermostated" equations of motion were developed. All were consistent with the canonical Gaussian distribution for the oscillator coordinate and momentum. The ergodicity of the various approaches has undergone several investigations, with somewhat inconclusive ( contradictory ) results. Here we illustrate several ways to test ergodicity and challenge the reader to find even more convincing algorithms or an entirely new approach to this problem.