Partial differential equations and stochastic methods in molecular dynamics*

Abstract

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.