Numerical Integration of the Extended Variable Generalized Langevin Equation with a Positive Prony Representable Memory Kernel

TL;DR

This paper develops a family of efficient, stable numerical integrators for the Generalized Langevin Equation with a positive Prony series memory kernel, suitable for molecular dynamics simulations involving complex viscoelastic and anomalous diffusive systems.

Contribution

It introduces a new class of extended variable integrators with stability and convergence analysis, implemented in LAMMPS, that conserve moments and avoid explicit history storage.

Findings

Demonstrates accuracy across canonical examples

Shows stability in the limit of Langevin dynamics

Maps confining potential effects onto a memory kernel

Abstract

Generalized Langevin dynamics (GLD) arise in the modeling of a number of systems, ranging from structured fluids that exhibit a viscoelastic mechanical response, to biological systems, and other media that exhibit anomalous diffusive phenomena. Molecular dynamics (MD) simulations that include GLD in conjunction with external and/or pairwise forces require the development of numerical integrators that are efficient, stable, and have known convergence properties. In this article, we derive a family of extended variable integrators for the Generalized Langevin equation (GLE) with a positive Prony series memory kernel. Using stability and error analysis, we identify a superlative choice of parameters and implement the corresponding numerical algorithm in the LAMMPS MD software package. Salient features of the algorithm include exact conservation of the first and second moments of the equilibrium velocity distribution in some important cases, stable behavior in the limit of conventional Langevin dynamics, and the use of a convolution-free formalism that obviates the need for explicit storage of the time history of particle velocities. Capability is demonstrated with respect to accuracy in numerous canonical examples, stability in certain limits, and an exemplary application in which the effect of a harmonic confining potential is mapped onto a memory kernel.