Linear Analysis of Stochastic Verlet-Type Integrators for Langevin Equations

TL;DR

This paper develops an analytical framework to evaluate stochastic Verlet integrators for Langevin equations, focusing on their accuracy in simulating diffusion, drift, and statistical sampling, and identifies the GJ integrators as the most reliable.

Contribution

It introduces closed-form expressions to assess integrator performance on key physical quantities, enabling comparison of multiple integrators over a range of time steps.

Findings

GJ integrators accurately simulate diffusion, drift, and Boltzmann distribution.

The framework applies to twelve integrators from past decades.

GJ integrators outperform others in high-quality thermodynamic simulations.

Abstract

We provide an analytical framework for analyzing the quality of stochastic Verlet-type integrators for simulating the Langevin equation. Focusing only on basic objective measures, we consider the ability of an integrator to correctly simulate two characteristic configurational quantities of transport, a) diffusion on a flat surface and b) drift on a tilted planar surface, as well as c) statistical sampling of a harmonic potential. For any stochastic Verlet-type integrator expressed in its configurational form, we develop closed form expressions to directly assess these three most basic quantities as a function of the applied time step. The applicability of the analysis is exemplified through twelve representative integrators developed over the past five decades, and algorithm performance is conveniently visualized through the three characteristic measures for each integrator. The GJ set of integrators stands out as the only option for correctly simulating diffusion, drift, and Boltzmann distribution in linear systems, and we therefore suggest that this general method is the one best suited for high quality thermodynamic simulations of nonlinear and complex systems, including for relatively high time steps compared to simulations with other integrators.