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.

本文介绍了使用Nosé– Hoover型恒温器的多维谐波振荡器系统的非遍历性的简单证明和详细分析。非遍历性的起源是多维目标物理系统中的对称性，它不同于带有一维谐波振荡器的Nosé– Hoover温控器中的对称性。还提出了一种新的简单的确定性方法来恢复遍历性。每个自由度都会附加一个单独的恒温器变量，并且所有这些变量都将作用于每个自由度的摩擦系数。此动作是线性的，由Nosé质量矩阵\（\ mathbf {Q} \）控制，它是标量Nosé质量的矩阵类似物。矩阵\（\ mathbf {Q} \） 可以打破对称性并有助于遍历。