Mathematical analysis and numerical methods for the computation of transport coefficients in molecular dynamics

TL;DR

This paper reviews numerical methods for computing transport coefficients in molecular dynamics, analyzing their accuracy, efficiency, and recent variance reduction techniques.

Contribution

It categorizes and analyzes three main classes of methods for calculating transport coefficients, including recent variance reduction strategies.

Findings

All three classes of methods have specific advantages and limitations.

Numerical errors can be estimated and controlled in these methods.

Variance reduction techniques improve computational efficiency.

Abstract

We review various numerical approaches to compute transport coefficients in molecular dynamics. These approaches can be broadly classified into three groups: (i) nonequilibrium methods based on applying an external driving field to the system, measuring the average response in the system, and evaluating the related linear response coefficient; (ii) approaches reformulating the transport coefficient of interest through a time correlation function for the equilibrium dynamics (the most popular instances being Green--Kubo and Einstein formulas); (iii) transient techniques, where the transport coefficient can be computed by monitoring the return to the steady state of a dynamics perturbed off its stationary distribution. For all three classes of methods, we provide elements of numerical analysis, allowing to estimate or at least quantify the level of numerical errors in the estimator of the transport coefficient; and also briefly present recent attempts to more efficiently compute transport coefficients with variance reduction approaches such as control variates, importance sampling and coupling methods. The computation of transport coefficients remains nonetheless challenging and will continue requiring research efforts in the foreseeable future.