Rational Construction of Stochastic Numerical Methods for Molecular Sampling

TL;DR

This paper develops a formal series expansion for Langevin dynamics integrators to improve molecular sampling accuracy, demonstrating superconvergence and significant efficiency gains in configurational sampling.

Contribution

It introduces a novel approach using the Baker-Campbell-Hausdorff lemma to analyze and compare Langevin integrators, revealing superconvergence properties and practical modifications for enhanced sampling.

Findings

Superconvergence (4th order accuracy) in high friction limit for one integrator.

Up to two orders of magnitude improvement in sampling accuracy.

Efficient methods requiring only one force evaluation per timestep.

Abstract

In this article, we focus on the sampling of the configurational Gibbs-Boltzmann distribution, that is, the calculation of averages of functions of the position coordinates of a molecular $N$-body system modelled at constant temperature. We show how a formal series expansion of the invariant measure of a Langevin dynamics numerical method can be obtained in a straightforward way using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics integrators in terms of their invariant distributions and demonstrate a superconvergence property (4th order accuracy where only 2nd order would be expected) of one method in the high friction limit; this method, moreover, can be reduced to a simple modification of the Euler-Maruyama method for Brownian dynamics involving a non-Markovian (coloured noise) random process. In the Brownian dynamics case, 2nd order accuracy of the invariant density is achieved. All methods considered are efficient for molecular applications (requiring one force evaluation per timestep) and of a simple form. In fully resolved (long run) molecular dynamics simulations, for our favoured method, we observe up to two orders of magnitude improvement in configurational sampling accuracy for given stepsize with no evident reduction in the size of the largest usable timestep compared to common alternative methods.