Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator

The Nosé-Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton's equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nosé-Hoover dynamics being ergodic. Hoover presented numerical experiments to show that the Nosé-Hoover dynamics are non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nosé-Hoover thermostat does not give an ergodynamical system for the one- dimensional harmonic oscillator when the "mass" of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nosé-Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass.