Fitting SDE models to nonlinear Kac–Zwanzig heat bath models

We study a class of "particle in a heat bath" models, which are a generalization of the well-known Kac–Zwanzig class of models, but where the coupling between the distinguished particle and the n heat bath particles is through nonlinear springs. The heat bath particles have random initial data drawn from an equilibrium Gibbs density. The primary objective is to approximate the forces exerted by the heat bath—which we do not want to resolve—by a stochastic process. By means of the central limit theorem for Gaussian processes, and heuristics based on linear response theory, we demonstrate conditions under which it is natural to expect that the trajectories of the distinguished particle can be weakly approximated, as n→∞, by the solution of a Markovian SDE. The quality of this approximation is verified by numerical calculations with parameters chosen according to the linear response theory. Alternatively, the parameters of the effective equation can be chosen using time series analysis. This is done and agreement with linear response theory is shown to be good.