A mathematical formalization of the parallel replica dynamics

TL;DR

This paper establishes the mathematical foundations of the parallel replica dynamics method, enabling more rigorous analysis and potential improvements in simulating stochastic processes in computational physics.

Contribution

It provides a formal mathematical framework for the parallel replica dynamics, connecting it with Markov process theory and quasi-stationary distributions.

Findings

The approach can achieve near-linear speed-up with the number of realizations.

A rigorous mathematical setting for performance assessment is developed.

Connections with Markov processes enable potential method improvements.

Abstract

The purpose of this article is to lay the mathematical foundations of a well known numerical approach in computational statistical physics and molecular dynamics, namely the parallel replica dynamics introduced by A.F. Voter. The aim of the approach is to efficiently generate a coarse-grained evolution (in terms of state-to-state dynamics) of a given stochastic process. The approach formally consists in concurrently considering several realizations of the stochastic process, and tracking among the realizations that which, the soonest, undergoes an important transition. Using specific properties of the dynamics generated, a computational speed-up is obtained. In the best cases, this speed-up approaches the number of realizations considered. By drawing connections with the theory of Markov processes and, in particular, exploiting the notion of quasi-stationary distribution, we provide a mathematical setting appropriate for assessing theoretically the performance of the approach, and possibly improving it.