Analysis and experiments for a computational model of a heat bath

A question of some interest in computational statistical mechanics is wheth er macroscopic quantities can be accurately computed without detailed resol ution of the fastest scales in the problem. To address this question a simp le model for a distinguished particle immersed in a heat bath is studied (d ue to Ford and Kac). The model yields a Hamiltonian system of dimension 2N + 2 for the distinguished particle and the degrees of freedom describing th e bath. It is proven that, in the limit of an infinite number of particles in the heat bath (N --> infinity), the motion of the distinguished particle is governed by a stochastic differential equation (SDE) of dimension 2. Nu merical experiments are then conducted on the Hamiltonian system of dimensi on 2N + 2 (N >> 1) to investigate whether the motion of the distinguished p article is accurately computed (i.e., whether it is close to the solution o f the SDE) when the time step is small relative to the natural time scale o f the distinguished particle, but the product of the fastest frequency in t he heat bath and the time step is not small-the underresolved regime in whi ch many computations are performed. It is shown that certain methods accura tely compute the limiting behavior of the distinguished particle, while oth ers do not. Those that do not are shown to compute a different, incorrect. macroscopic limit.