Singly-Thermostated Ergodicity in Gibbs' Canonical Ensemble and the 2016 Ian Snook Prize

TL;DR

This paper reviews progress in developing single-thermostat algorithms for ergodic sampling of Gibbs' canonical ensemble, highlighting recent advances and remaining challenges for anharmonic potentials.

Contribution

It summarizes three key steps in improving Nosé's thermostat approach and identifies the unresolved problem of ergodic sampling for anharmonic systems.

Findings

Multiple two-thermostat sets achieve ergodicity in phase space.

Single-thermostat weak control methods are ergodic for harmonic systems.

Ergodic sampling for anharmonic potentials remains unsolved.

Abstract

For a harmonic oscillator, Nos\'e's single-thermostat approach to simulating Gibbs' canonical ensemble with dynamics samples only a small fraction of the phase space. Nos\'e's approach has been improved in a series of three steps: [ 1 ] several two-thermostat sets of motion equations have been found which cover the complete phase space in an ergodic fashion, [ 2 ] sets of single-thermostat motion equations, exerting "weak control" over both forces and momenta, have been shown to be ergodic, and [ 3 ] sets of single-thermostat motion equations exerting weak control over two velocity moments provide ergodic phase-space sampling for the oscillator and for the rigid pendulum, but not for the quartic oscillator or for the Mexican Hat potential. The missing fourth step, motion equations providing ergodic sampling for anharmonic potentials requires a further advance. The 2016 Ian Snook Prize will be awarded to the author(s) of the most interesting original submission addressing the problem of finding ergodic algorithms for Gibbs' canonical ensemble using a single thermostat.