Two-time-scale Relaxation towards Thermal Equilibrium of the Enigmatic Piston

We investigate the evolution of a system composed of N non-interacting point particles of mass m in a container divided into two chambers by a movable adiabatic piston of mass M ≫ m. Using a two-time-scale perturbation approach in terms of the small parameter α = 2m/(M +m), we show that the evolution towards thermal equilibrium proceeds in two stages. The first stage is a fast, deterministic, adiabatic relaxation towards mechanical equilibrium. The second stage, which takes place at times O(M), is a slow fluctuation-driven, diathermic relaxation towards thermal equilibrium. A very simple equation is derived which shows that in the second stage, the position of the piston is given by XM (t) = L[1/2 − ξ(αt)] where the function ξ is independent of M . Numerical simulations support the assumptions underlying our analytical derivations and illustrate the large mass range in which the picture holds.