On Ergodicity for Multidimensional Harmonic Oscillator Systems with Nosé – Hoover-type Thermostat

A simple proof and a detailed analysis of the nonergodicity for multidimensional harmonic oscillator systems with the Nosé – Hoover-type thermostat are presented. The origin of the nonergodicity is the symmetries in the multidimensional target physical system, and it differs from that in the Nosé – Hoover thermostat with the one-dimensional harmonic oscillator. A new and simple deterministic method to recover the ergodicity is also proposed. An individual thermostat variable is attached to each degree of freedom, and all of these variables act on a friction coefficient for each degree of freedom. This action is linear and controlled by a Nosé mass matrix $\mathbf{Q}$, which is the matrix analogue of the scalar Nosé mass. The matrix $\mathbf{Q}$ can break the symmetry and contribute to the ergodicity.