Diffusion modeling of the Rayleigh piston

A Markov jump process in which a massive labeled particle undergoes random elastic collisions with a thermal bath is investigated. It is found that the behavior of the labeled particle can be divided into three distinct regimes depending on whether its velocity is (1) much less than, (2) on the order of, or (3) much greater than the mean speed of a bath particle. In each regime the jump process can be approximated by a particular continuous-path diffusion process. The first case corresponds to the Ornstein-Uhlenbeck process, while each of the latter can be modeled by a deterministic process with a nonlinear Langevin equation. In addition, in cases (2) and (3), the scaled deviation from the mean velocity can be modeled by a nonstationary diffusion. By scaling the time and letting the mass of the labeled particle become large, a continuous-path diffusion is constructed which approximates the jump process in each regime. Analytic solutions for the transition probability density are provided in each case, and numerical comparisons are made between the mean and variance of the diffusions and the original jump process.