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

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 nonergodicity is symmetries in target physical system, differ from that 1-dimensional oscillator. new deterministic method to recover ergodicity also presented. An individual variable attached each degree freedom, all these variables act a friction coefficient freedom. This action linear controlled by Nos\'e mass matrix Q, which analogue scalar Nos\'e's mass. Matrix Q can break symmetry contribute attain ergodicity.