Nose-Hoover恒温热浴谐振子的非遍历性
Non-ergodicity of the Nose-Hoover Thermostatted Harmonic Oscillator
AI总结：
本文利用KAM理论证明当储层质量较大时，Nose-Hoover恒温器对一维谐振子不产生遍历动力学，并通过数值实验支持该结论。
摘要：
Nose-Hoover恒温器是一种确定性动力系统，旨在计算正则吉布斯分布的相空间积分。牛顿方程通过将额外的储层变量耦合到物理变量进行修改。根据吉布斯测度正确采样相空间依赖于Nose-Hoover动力学的遍历性。Hoover展示了数值实验，表明Nose-Hoover动力学在应用于谐振子时是非遍历的。在本文中，我们证明当储层的“质量”较大时，Nose-Hoover恒温器对一维谐振子不产生遍历动力学。我们的非遍历性证明使用KAM理论来证明Nose-Hoover动力系统存在不变环面，将相空间分隔成不变区域。我们提出了由我们的分析启发的数值实验，似乎表明即使对于中等恒温器质量，动力学也不是遍历的。我们还给出了将两个恒温器应用于一维谐振子的Nose-Hoover链的数值实验。这些实验似乎支持如果储层质量足够大则动力学非遍历，而对于更中等质量则与遍历性一致。
英文摘要：
The Nose-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 Nose-Hoover dynamics being ergodic. Hoover presented numerical experiments that show the Nose-Hoover dynamics to be non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nose-Hoover thermostat does not give an ergodic dynamics 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 Nose-Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamics is not ergodic even for a moderate thermostat mass. We also give numerical experiments of the Nose-Hoover chain with two thermostats applied to the one-dimensional harmonic oscillator. These experiments seem to support the non-ergodicity of the dynamics if the masses of the reservoirs are large enough and are consistent with ergodicity for more moderate masses.
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