Molecular dynamics and the accuracy of numerically computed averages

Molecular dynamics is discussed from a mathematical perspective. The recent history of method development is briefly surveyed with an emphasis on the use of geometric integration as a guiding principle. The recovery of statistical mechanical averages from molecular dynamics is then introduced, and the use of backward error analysis as a technique for analysing the accuracy of numerical averages is described. This article gives the first rigorous estimates for the error in statistical averages computed from molecular dynamics simulation based on backward error analysis. It is shown that molecular dynamics introduces an appreciable bias at stepsizes which are below the stability threshold. Simulations performed in such a regime can be corrected by use of a stepsize-dependent reweighting factor. Numerical experiments illustrate the efficacy of this approach. In the final section, several open problems in dynamics-based molecular sampling are considered.