Linear Analysis of Stochastic Verlet-Type Integrators for Langevin Equations

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.

Latest version corrects two inconsequential typographical errors of previous versions: 1) Equation reference just above Eq. (9), and 2) numerator corrected in last term of Eq. (50a)

References in corpus (15)

• A Dynamical Approach to Temperature
• A simple and effective Verlet-type algorithm for simulating Langevin dynamics
• ESPResSo 4.0 -- An Extensible Software Package for Simulating Soft Matter Systems
• RUMD: A general purpose molecular dynamics package optimized to utilize GPU hardware down to a few thousand particles
• Complete Set of Stochastic Verlet-Type Thermostats for Correct Langevin Simulations
• Time step rescaling recovers continuous-time dynamical properties for discrete-time Langevin integration of nonequilibrium systems
• Trotter Derivation of Algorithms for Brownian and Dissipative Particle Dynamics
• Accurate Configurational and Kinetic Statistics in Discrete-Time Langevin Systems
• Comparison of Modern Langevin Integrators for Simulations of Coarse-Grained Polymer Melts
• Configurational temperature in active matter. I. Lines of invariant physics in the phase diagram of the Ornstein-Uhlenbeck model
• On the Application of Non-Gaussian Noise in Stochastic Langevin Simulations
• The Challenge of Stochastic Størmer-Verlet Thermostats Generating Correct Statistics
• Bringing discrete-time Langevin splitting methods into agreement with thermodynamics
• Comparison of effective and stable Langevin dynamics integrators
• On the Definition of Velocity in Discrete-Time, Stochastic Langevin Simulations