Journal of Statistical Physics

, Volume 112, Issue 5–6, pp 1177–1206 | Cite as

Two-Time-Scale Relaxation Towards Thermal Equilibrium of the Enigmatic Piston

  • Christian Gruber
  • Séverine Pache
  • Annick Lesne
Article

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 Mm. 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 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.

Liouville equation adiabatic mechanical equilibrium thermal equilibrium perturbation 

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Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Christian Gruber
    • 1
  • Séverine Pache
    • 1
  • Annick Lesne
    • 2
  1. 1.École Polytechnique Fédérale de LausanneInstitut de Physique ThéoriqueLausanneSwitzerland
  2. 2.Laboratoire de Physique Théorique des LiquidesUniversité Pierre et Marie Curie, Case 121Paris Cedex 05France

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