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

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.