On ergodicity for multi-dimensional harmonic oscillator systems with Nose-Hoover type thermostat

TL;DR

This paper analyzes why multidimensional harmonic oscillators with Nose-Hoover thermostats are non-ergodic due to symmetries, and proposes a new method attaching individual thermostats to restore ergodicity by breaking these symmetries.

Contribution

It provides a simple proof of non-ergodicity in multidimensional systems and introduces a novel deterministic approach using a matrix Nosé mass to recover ergodicity.

Findings

Identifies symmetry as the cause of non-ergodicity in multidimensional systems.

Proposes a new thermostat method with individual variables for each degree of freedom.

Demonstrates that the matrix Nosé mass can break symmetries and achieve ergodicity.

Abstract

A simple proof and detailed analysis on the non-ergodicity for multidimensional harmonic oscillator systems with Nose-Hoover type thermostat are given. The origin of the nonergodicity is symmetries in the multidimensional target physical system, and is differ from that in the Nose-Hoover thermostat with the 1-dimensional harmonic oscillator. A new simple deterministic method to recover the ergodicity is also presented. An individual thermostat variable is attached to each degree of freedom, and all these variables act on a friction coefficient for each degree of freedom. This action is linear and controlled by a Nos\'e mass matrix Q, which is a matrix analogue of the scalar Nos\'e's mass. Matrix Q can break the symmetry and contribute to attain the ergodicity.