Complete Set of Stochastic Verlet-Type Thermostats for Correct Langevin Simulations

We present the complete set of stochastic Verlet-type algorithms that can provide correct statistical measures for both configurational and kinetic sampling in discrete-time Langevin systems. The approach is a brute-force general representation of the Verlet-algorithm with free parameter coefficients that are determined by requiring correct Boltzmann sampling for linear systems, regardless of time step. The result is a set of statistically correct methods given by one free functional parameter, which can be interpreted as the one-time-step velocity attenuation factor. We define the statistical characteristics of both true on-site $v^n$ and true half-step $u^{n+frac{1}{2}}$ velocities, and use these definitions for each statistically correct Stormer-Verlet method to find a unique associated half-step velocity expression, which yields correct kinetic Maxwell-Boltzmann statistics for linear systems. It is shown that no other similar, statistically correct on-site velocity exists. We further discuss the use and features of finite-difference velocity definitions that are neither true on-site, nor true half-step. The set of methods is written in convenient and conventional stochastic Verlet forms that lend themselves to direct implementation for, e.g., Molecular Dynamics applications. We highlight a few specific examples, and validate the algorithms through comprehensive Langevin simulations of both simple nonlinear oscillators and complex Molecular Dynamics.