We expand on two previous developments in the modeling of discrete-time Langevin systems. One is the well-documented Grønbech-Jensen Farago (GJF) thermostat, which has been demonstrated to give robust and accurate configurational sampling of the phase space. Another is the recent discovery that also kinetics can be accurately sampled for the GJF method. Through a complete investigation of all possible finite-difference approximations to the velocity, we arrive at two main conclusions: (1) It is not possible to define a so-called on-site velocity such that kinetic temperature will be correct and independent of the time step, and (2) there exists a set of infinitely many possibilities for defining a two-point (leap-frog) velocity that measures kinetic energy correctly for linear systems in addition to the correct configurational statistics obtained from the GJF algorithm. We give explicit expressions for the possible definitions, and we incorporate these into convenient and practical algorithmic forms of the normal Verlet-type algorithms along with a set of suggested criteria for selecting a useful definition of velocity.

中文翻译：

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在Grønbech-JensenFarago温控器中定义速度以进行准确的动力学统计

我们在离散时间Langevin系统的建模中扩展了先前的两个开发。一个是有据可查的Grønbech-JensenFarago（GJF）温控器，它已被证明可以对相空间进行稳健而准确的配置采样。另一个是最近的发现，即对于GJF方法也可以准确地采样动力学。通过对速度的所有可能的有限差分近似的全面研究，我们得出两个主要结论：（1）不是可以定义一个所谓的现场速度，以使运动温度是正确的并且与时间步长无关，并且（2）存在用于定义两点（跳跃青蛙）速度的无限多种可能性除了从GJF算法获得的正确配置统计信息外，还可以为线性系统正确测量动能。我们为可能的定义给出了明确的表达式，并将它们结合到普通Verlet型算法的方便实用的算法形式中，并结合了一组建议的标准，用于选择有用的速度定义。