时间可逆恒温谐振子的遍历性和2014年Ian Snook奖
Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator and the 2014 Ian Snook Prize
摘要：
1984年，野濑修一开创了原子模拟的新世界。他制定了一个汉密尔顿量身定制生成吉布斯的正则分布动态。这个聪明的想法在微正则分子动力学和正则统计力学之间架起了一座桥梁。在此之前，正则分布是用蒙特卡罗抽样来研究的。诺斯的动力学哈密顿桥需要空间填充结构的“遍历”支持，以重现整个分布。对于足够小的系统，如谐振子，诺斯的动力学方法不符合吉布斯的采样，而是显示了一个复杂的结构，分为混沌海，岛屿和岛屿链，这是熟悉的教科书从汉密尔顿混沌的调查。在试图提高小系统遍历性几个更复杂的“恒温”运动方程的开发。所有的振子坐标和动量都符合正则高斯分布。各种方法的遍历性经历了几次调查，结果有些不确定（矛盾）。在这里，我们举例说明几种方法来测试遍历性和挑战读者找到更令人信服的算法或一个全新的方法来解决这个问题。
英文摘要：
Shuichi Nosé opened up a new world of atomistic simulation in 1984. He formulated a Hamiltonian tailored to generate Gibbs' canonical distribution dynamically. This clever idea bridged the gap between microcanonical molecular dynamics and canonical statistical mechanics. Until then the canonical distribution was explored with Monte Carlo sampling. Nosé's dynamical Hamiltonian bridge requires the "ergodic" support of a space-filling structure in order to reproduce the entire distribution. For sufficiently small systems, such as the harmonic oscillator, Nosé's dynamical approach failed to agree with Gibbs' sampling and instead showed a complex structure, partitioned into a chaotic sea, islands, and chains of islands, that is familiar textbook fare from investigations of Hamiltonian chaos. In trying to enhance small-system ergodicity several more complicated "thermostated" equations of motion were developed. All were consistent with the canonical Gaussian distribution for the oscillator coordinate and momentum. The ergodicity of the various approaches has undergone several investigations, with somewhat inconclusive ( contradictory ) results. Here we illustrate several ways to test ergodicity and challenge the reader to find even more convincing algorithms or an entirely new approach to this problem.
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