Ergodicity of Thermostat Family of Nos\'e--Hoover type

TL;DR

This paper explores a family of thermostats generalizing Nosé–Hoover, deriving conditions for their equations of motion, and demonstrating that single-variable thermostats may lose ergodicity, implying the need for multi-variable thermostats.

Contribution

It derives a partial differential equation condition for thermostat equations and introduces a family of thermostats including Nosé–Hoover as a minimal solution.

Findings

Single-variable thermostats can lose ergodicity with large relaxation times.

The family of thermostats includes the Nosé–Hoover method.

Multi-variable thermostats are necessary to ensure ergodicity.

Abstract

One-variable thermostats are studied as a generalization of the Nos\'e--Hoover method which is aimed to achieve Gibbs' canonical distribution with conserving the time-reversibility. A condition for equations of motion for the system with the thermostats is derived in the form of a partial differential equation. Solutions of this equation construct a family of thermostats including the Nos\'e--Hoover method as the minimal solution. It is shown that the one-variable thermostat coupled with the one-dimensional harmonic oscillator loses its ergodicity with large enough relaxation time. The present result suggests that multi-variable thermostats are required to assure the ergodicity and to work as heatbath.