Two-time-scale relaxation toward thermal equilibrium of the enigmatic piston

C. Gruber (1) , S. Pache (1) , A. Lesne (2)

1 Institut de Théorie des Phénomènes Physiques
2 LPTL - Laboratoire de Physique Théorique des Liquides

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 $\alpha=2m/(M+m)$, we show that the evolution towards thermal equilibrium proceeds in two stages. The first stage %, at time ${\cal O}(1)$, is a fast, deterministic, adiabatic relaxation towards mechanical equilibrium. The second stage, which takes place at times ${\cal O}(M)$, 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 $X_M(t)=L[1/2-\xi(\alpha t)]$ 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.

Mots clés: Liouville equation; Adiabatic; Mechanical equilibrium; Thermal equilibrium; Perturbation

Domaines: Mécanique statistique [cond-mat.stat-mech]

C. Gruber, S. Pache, A. Lesne. Two-time-scale relaxation toward thermal equilibrium of the enigmatic piston. Journal of Statistical Physics, 2003, 112, pp.1199-1228. ⟨hal-00003836⟩