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

TL;DR

This paper introduces a complete set of stochastic Verlet algorithms that ensure correct statistical sampling in Langevin simulations, applicable to molecular dynamics, by defining velocities and parameters that yield accurate thermodynamic properties.

Contribution

It provides a comprehensive family of Verlet-type thermostats with a free parameter for correct Boltzmann sampling, including velocity definitions that guarantee accurate kinetic statistics in Langevin systems.

Findings

The algorithms produce correct configurational and kinetic sampling.

Validation through Langevin simulations of oscillators and molecular systems.

Unique velocity expressions ensure proper Maxwell-Boltzmann statistics.

Abstract

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.