Rational Construction of Stochastic Numerical Methods for Molecular Sampling

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 modeled at constant temperature. Using the Baker-Campbell-Hausdorff expansion, we compare Langevin dynamics integrators in terms of their invariant distributions and demonstrate a superconvergence property of a certain 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 fully resolved (long run) molecular dynamics simulations, for our favored method, we observe up to two orders of magnitude improvement in the configurational sampling accuracy for given stepsize with no evident reduction in the size of the largest usable timestep compared with common alternatives.