Ergodicity of the Martyna-Klein-Tuckerman Thermostat and the 2014 Snook Prize

TL;DR

This paper investigates the ergodicity of the Martyna-Klein-Tuckerman thermostat, providing new insights into its chaotic dynamics and demonstrating its compatibility with multiple ergodicity tests, thus supporting its effectiveness in generating Gibbs' canonical distribution.

Contribution

The paper offers a detailed analysis of the ergodicity of the MKT thermostat, including new results on its chaotic behavior near fixed points and comprehensive testing for ergodicity.

Findings

MKT equations are ergodic according to six tests

Chaotic dynamics observed near fixed points

Supports MKT's suitability for canonical ensemble simulations

Abstract

Nos\'e and Hoover's 1984 work showed that although Nos\'e and Nos\'e-Hoover dynamics were both consistent with Gibbs' canonical distribution neither dynamics, when applied to the harmonic oscillator, provided Gibbs' Gaussian distribution. Further investigations indicated that two independent thermostat variables are necessary, and often sufficient, to generate Gibbs' canonical distribution for an oscillator. Three successful time-reversible and deterministic sets of two-thermostat motion equations were developed in the 1990s. We analyze one of them here. It was developed by Martyna, Klein, and Tuckerman in 1992. Its ergodicity was called into question by Patra and Bhattacharya in 2014. This question became the subject of the 2014 Snook Prize. Here we summarize the previous work on this problem and elucidate new details of the chaotic dynamics in the neighborhood of the two fixed points. We apply six separate tests for ergodicity and conclude that the MKT equations are fully compatible with all of them, in consonance with our recent work with Clint Sprott and Puneet Patra.