  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 a la Talay-Tubaro
on the invariant distribution for small stepsize, and compare the sampling bias
obtained for various choices of 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.
