Nose-Hoover动力学的非遍历性
Non-ergodicity of Nose-Hoover dynamics
AI总结：
本文证明了Nose-Hoover动力学在一维非谐振子系统以及某些多维系统中的非遍历性，并通过数值实验验证了一维摆问题和二维中心力问题的非遍历性。
摘要：
Nose-Hoover动力学的数值积分提供了一种确定性方法，用于采样正则吉布斯测度。Nose-Hoover动力学通过添加一个与物理变量非线性耦合的“热浴”变量，扩展了物理哈密顿动力学。该方法的准确性取决于动力学的遍历性。早期的数值实验表明，对于某些模型问题，动力学存在非遍历性。作者最近证明了一维谐振子的Nose-Hoover动力学是非遍历的。本文将此结果推广到非谐振的一维系统。还表明，对于某些多维系统，在热浴“质量”趋于无穷大的极限下，平均动力学具有多个不变量，从而为动力学非遍历性或慢遍历化提供了理论支持。对于二维中心力问题和一维摆问题的数值实验给出了非遍历性的证据。
英文摘要：
The numerical integration of the Nose-Hoover dynamics gives a deterministic method that is used to sample the canonical Gibbs measure. The Nose-Hoover dynamics extends the physical Hamiltonian dynamics by the addition of a "thermostat" variable, that is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nose-Hoover dynamics for the one-dimensional harmonic oscillator. In this paper, this result is extended to non-harmonic one-dimensional systems. It is also shown for some multidimensional systems that the averaged dynamics for the limit of infinite thermostat "mass" have many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity.
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