Computational statistical physics and hypocoercivity

TL;DR

This paper introduces molecular dynamics within statistical physics, focusing on Langevin dynamics and hypocoercivity techniques to analyze long-term convergence and error estimates in molecular simulations.

Contribution

It reviews hypocoercive methods for analyzing Langevin dynamics and their application to error estimation in molecular property computations.

Findings

Hypocoercivity techniques ensure convergence of Langevin dynamics.

Error estimates for molecular averages are derived from asymptotic variance.

Long-term behavior of stochastic processes is characterized using degenerate elliptic operators.

Abstract

This note provides an introduction to molecular dynamics, the computational implementation of the theory of statistical physics. The discussion is focused on the properties of Langevin dynamics, a degenerate stochastic differential equation which can be seen as a perturbation of Hamiltonian dynamics. From an analytical point of view, the generator of Langevin dynamics is a degenerate elliptic operator. The evolution of the law of the stochastic process is governed by the Fokker-Planck equation, and its longtime convergence can be obtained via hypocoercive techniques, some of which are reviewed here. One consequence of these analytical results in terms of error estimates for the computation of average properties of molecular systems is the estimation of the asymptotic variance of time averages in a central limit theorem.