Abstract
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 \(\mathcal{O}\)(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[1/2−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.
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Gruber, C., Pache, S. & Lesne, A. Two-Time-Scale Relaxation Towards Thermal Equilibrium of the Enigmatic Piston. Journal of Statistical Physics 112, 1177–1206 (2003). https://doi.org/10.1023/A:1024671710343
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DOI: https://doi.org/10.1023/A:1024671710343