The objective of molecular dynamics computations is to infer macroscopic properties of matter from atomistic models via averages with respect to probability measures dictated by the principles of statistical physics. Obtaining accurate results requires efficient sampling of atomistic configurations, which are typically generated using very long trajectories of stochastic differential equations in high dimensions, such as Langevin dynamics and its overdamped limit. Depending on the quantities of interest at the macroscopic level, one may also be interested in dynamical properties computed from averages over paths of these dynamics.This review describes how techniques from the analysis of partial differential equations can be used to devise good algorithms and to quantify their efficiency and accuracy. In particular, a crucial role is played by the study of the long-time behaviour of the solution to the Fokker–Planck equation associated with the stochastic dynamics.

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 estimates à la Talay–Tubaro on the invariant distribution for small stepsize, and compare the sampling bias obtained for various choices of the 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.

The purpose of the present article is to compare different phase-space sampling methods, such as purely stochastic methods (Rejection method, Metropolized independence sampler, Importance Sampling), stochastically perturbed Molecular Dynamics methods (Hybrid Monte Carlo, Langevin Dynamics, Biased Random Walk), and purely deterministic methods (Nosé-Hoover chains, Nosé-Poincaré and Recursive Multiple Thermostats (RMT) methods). After recalling some theoretical convergence properties for the various methods, we provide some new convergence results for the Hybrid Monte Carlo scheme, requiring weaker (and easier to check) conditions than previously known conditions. We then turn to the numerical efficiency of the sampling schemes for a benchmark model of linear alkane molecules. In particular, the numerical distributions that are generated are compared in a systematic way, on the basis of some quantitative convergence indicators.

In this paper, we consider Langevin processes with mechanical constraints. The latter are a fundamental tool in molecular dynamics simulation for sampling purposes and for the computation of free energy differences. The results of this paper can be divided into three parts. (i) We propose a simple discretization of the constrained Langevin process based on a splitting strategy. We show how to correct the scheme so that it samples exactly the canonical measure restricted on a submanifold, using a Metropolis-Hastings correction in the spirit of the Generalized Hybrid Monte Carlo (GHMC) algorithm. Moreover, we obtain, in some limiting regime, a consistent discretization of the overdamped Langevin (Brownian) dynamics on a submanifold, also sampling exactly the correct canonical measure with constraints. (ii) For free energy computation using thermodynamic integration, we rigorously prove that the longtime average of the Lagrange multipliers of the constrained Langevin dynamics yields the gradient of a rigid version of the free energy associated with the constraints. A second order time discretization using the Lagrange multipliers is proposed. (iii) The Jarzynski-Crooks fluctuation relation is proved for Langevin processes with mechanical constraints evolving in time. An original numerical discretization without time discretization error is proposed, and its overdamped limit is studied. Numerical illustrations are provided for (ii) and (iii).