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.

We consider numerical methods for thermodynamic sampling, i.e., computing sequences of points distributed according to the Gibbs–Boltzmann distribution, using Langevin dynamics and overdamped Langevin dynamics (Brownian dynamics). A wide variety of numerical methods for Langevin dynamics may be constructed based on splitting the stochastic differential equations into various component parts, each of which may be propagated exactly in the sense of distributions. Each such method may be viewed as generating samples according to an associated invariant measure that differs from the exact canonical invariant measure by a stepsize-dependent perturbation. We provide error estimates à la Talay–Tubaro on the invariant distribution for small stepsize, and compare the sampling bias obtained for various choices of the splitting method. We further investigate the overdamped limit and apply the methods in the context of driven systems where the goal is sampling with respect to a nonequilibrium steady state. Our analyses are illustrated by numerical experiments.

We describe parallel Markov chain Monte Carlo methods that propagate a collective ensemble of paths, with local covariance information calculated from neighbouring replicas. The use of collective dynamics eliminates multiplicative noise and stabilizes the dynamics, thus providing a practical approach to difficult anisotropic sampling problems in high dimensions. Numerical experiments with model problems demonstrate that dramatic potential speedups, compared to various alternative schemes, are attainable.

Langevin dynamics is a versatile stochastic model used in biology, chemistry, engineering, physics and computer science. Traditionally, in thermal equilibrium, one assumes (i) the forces are given as the gradient of a potential and (ii) a fluctuation-dissipation relation holds between stochastic and dissipative forces; these assumptions ensure that the system samples a prescribed invariant Gibbs-Boltzmann distribution for a specified target temperature. In this article, we relax these assumptions, incorporating variable friction and temperature parameters and allowing nonconservative force fields, for which the form of the stationary state is typically not known a priori. We examine theoretical issues such as stability of the steady state and ergodic properties, as well as practical aspects such as the design of numerical methods for stochastic particle models. Applications to nonequilibrium systems with thermal gradients and active particles are discussed.