随机Størmer-Verlet恒温器产生正确统计数据的挑战
The Challenge of Stochastic Størmer-Verlet Thermostats Generating Correct Statistics
摘要：
鉴于最近开发的完整GJ集的单随机变量随机，离散时间Størmer-Verlet算法的统计精确模拟的朗之万方程，我们调查两个突出的问题：1）每个时间步包括多个随机变量是否有任何算法或统计学上的好处，以及2）使用可用的统计正确算法集中的一种或多种方法是否有客观原因？为了解决第一个问题，我们假设一个一般形式的离散时间方程有两个随机变量，然后遵循系统的，蛮力GJ方法，通过在线性系统中执行正确的热力学。它的结论是，正确的配置玻尔兹曼采样的一个粒子在一个谐波势意味着正确的配置自由粒子扩散，这些要求只能完成，如果每个时间步长的两个随机变量是相同的。因此，我们认为GJ集代表了所有可能的随机Størmer-Verlet方法，这些方法可以重现线性系统的时间步长无关统计。因此，第二个问题在GJ集合中得到解决。部分基于复杂分子系统的数值模拟，部分基于时间的解析标度，我们分析了不同方法之间稳定性的明显差异。我们将这种差异归因于每种方法中固有的时间缩放，并建议这种缩放可能导致动力学和统计模拟结果的解释不一致。因此，我们建议，最少的固有时间尺度的方法，GJ-I/GJF-2GJ方法，是首选的统计应用程序，虚假的时间尺度是不可取的。
英文摘要：
In light of the recently developed complete GJ set of single random variable stochastic, discrete-time Størmer-Verlet algorithms for statistically accurate simulations of Langevin equations, we investigate two outstanding questions: 1) Are there any algorithmic or statistical benefits from including multiple random variables per time-step, and 2) are there objective reasons for using one or more methods from the available set of statistically correct algorithms? To address the first question, we assume a general form for the discrete-time equations with two random variables and then follow the systematic, brute-force GJ methodology by enforcing correct thermodynamics in linear systems. It is concluded that correct configurational Boltzmann sampling of a particle in a harmonic potential implies correct configurational free-particle diffusion, and that these requirements only can be accomplished if the two random variables per time step are identical. We consequently submit that the GJ set represents all possible stochastic Størmer-Verlet methods that can reproduce time-step-independent statistics of linear systems. The second question is thus addressed within the GJ set. Based in part on numerical simulations of complex molecular systems, and in part on analytic scaling of time, we analyze the apparent difference in stability between different methods. We attribute this difference to the inherent time scaling in each method, and suggest that this scaling may lead to inconsistencies in the interpretation of dynamical and statistical simulation results. We therefore suggest that the method with the least inherent time-scaling, the GJ-I/GJF-2GJ method, be preferred for statistical applications where spurious rescaling of time is undesirable.
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