The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics

Abstract

We consider numerical methods for thermodynamic sampling, i.e. computing sequences of points distributed according to the Gibbs-Boltzmann distribution, using Langevin dynamics and overdamped Langevin dynamics (Brownian dynamics). A wide variety of numerical methods for Langevin dynamics may be constructed based on splitting the stochastic differential equations into various component parts, each of which may be propagated exactly in the sense of distributions. Each such method may be viewed as generating samples according to an associated invariant measure that differs from the exact canonical invariant measure by a stepsize-dependent perturbation. We provide error estimatesà la Talay-Tubaro on the invariant distribution for small stepsize, and compare the sampling bias obtained for various choices of splitting method. We further investigate the overdamped limit and apply the methods in the context of driven systems where the goal is sampling with respect to a nonequilibrium steady state. Our analyses are illustrated by numerical experiments.

Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

“…It is clear that the sum of the right-hand sides of these two systems gives equation (1.5). There are a number of ways of combining integrators of equations (4.1) [23]. We now define specific integrators for equations (4.1) and (4.2).…”

Langevin Dynamics with Variable Coefficients and Nonconservative Forces: From Stationary States to Numerical Methods

“…A sufficient condition for a central limit theorem of the form (15) to hold for the solution x of (5) and the observable ϕ is that the Poisson equation…”

“…Following [15], we break the Langevin system (1) and (2) into three parts: A and B corresponding to the deterministic flow:…”

“…Even for a particular choice of decomposition, there are many ways to combine the sequence of steps. The properties of various choices of the splitting method have been examined in detail for constant Γ and Σ in [15]. In that work it has been shown that the numerical methods constructed using certain components inherit the ergodicity properties of the SDE system, whereas the invariant measure of the numerical method differs from that of the SDE.…”

“…In the methods of [15,16] this is done exactly, however in the current setting the solution of a Lyapunov equation would need to be obtained at each step in time, a calculation that would dominate the computational load in a large scale simulation. Therefore, we offer an efficient numerical procedure based on multiple timestepping [17], relying on a further splitting of the OU equation at each interior timestep.…”