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

*2021, Volume 26, Number 2, pp. 183-204*

Author(s):

**Fukuda I., Moritsugu K., Fukunishi Y.**

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.

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