A simple and effective Verlet-type algorithm for simulating Langevin dynamics

TL;DR

This paper introduces a simple Verlet-type algorithm for Langevin dynamics that accurately reproduces diffusive behavior and Boltzmann distribution, suitable for molecular simulations with damping and constraints.

Contribution

A revised Verlet algorithm that correctly incorporates friction and noise, providing exact statistical properties across damping regimes.

Findings

Accurately reproduces diffusive behavior in flat potentials.

Provides exact Boltzmann distribution for harmonic oscillators.

Applicable to molecular dynamics with constraints.

Abstract

We present a revision to the well known Stormer-Verlet algorithm for simulating second order differential equations. The revision addresses the inclusion of linear friction with associated stochastic noise, and we analytically demonstrate that the new algorithm correctly reproduces diffusive behavior of a particle in a flat potential. For a harmonic oscillator, our algorithm provides the exact Boltzmann distribution for any value of damping, frequency, and time step for both underdamped and over damped behavior within the usual the stability limit of the Verlet algorithm. Given the structure and simplicity of the method we conclude this approach can trivially be adapted for contemporary applications, including molecular dynamics with extensions such as molecular constraints.