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 $M\gg m$. Using a two-time-scale perturbation approach in terms of the small parameter $\alpha =2m\mathrm{/}(M+m)$, 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 $\mathcal{O}(M)$, 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 $X_M(t)=L[1\mathrm{/}2-\xi (\alpha t)]$ where the function $\xi$ is independent of $M$. Numerical simulations support the assumptions underlying our analytical derivations and illustrate the large mass range in which the picture holds.