Non-ergodicity of Nose-Hoover dynamics

TL;DR

This paper investigates the non-ergodic behavior of Nose-Hoover dynamics, demonstrating through theoretical proofs and numerical experiments that it fails to sample the canonical distribution effectively in various systems.

Contribution

The authors extend previous non-ergodicity results from harmonic oscillators to more general one-dimensional and multidimensional systems, providing theoretical and numerical evidence.

Findings

Nose-Hoover dynamics is non-ergodic for certain systems.

Theoretical proof of non-ergodicity in non-harmonic 1D systems.

Numerical experiments support non-ergodicity in 2D and pendulum models.

Abstract

The numerical integration of the Nose-Hoover dynamics gives a deterministic method that is used to sample the canonical Gibbs measure. The Nose-Hoover dynamics extends the physical Hamiltonian dynamics by the addition of a "thermostat" variable, that is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nose-Hoover dynamics for the one-dimensional harmonic oscillator. In this paper, this result is extended to non-harmonic one-dimensional systems. It is also shown for some multidimensional systems that the averaged dynamics for the limit of infinite thermostat "mass" have many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity.