Smoluchowski-Kramers Approximation for Stochastic Differential Equations under Discretization

This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field. We show that the approximation holds for the drift-implicit Euler–Maruyama discretization and derive its convergence rate. In particular, the solution of the discretized system converges to the solution of the first-order limit equation in the mean-square sense, and this convergence is independent of the order in which the mass parameter μ and the step size h tend to zero.