Complete set of stochastic Verlet-type thermostats for correct Langevin simulations

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 $u^{n+1/2}$ velocities, and use these definitions for each statistically correct Størmer-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.

GRAPHICAL ABSTRACT

KEYWORDS:

• Molecular dynamics
• discrete-time Langevin equations
• Størmer-Verlet algorithms
• computational statistical mechanics
• stochastic thermostats

Acknowledgements

The author is grateful for encouraging discussions with Aidan Thompson at the outset of this work, and for conversations with Oded Farago and Richard Scalettar.

Disclosure statement

No potential conflict of interest was reported by the author.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.