On the adiabatic properties of a stochastic adiabatic wall: Evolution, stationary non-equilibrium, and equilibrium states

TL;DR

This paper investigates the stochastic adiabatic wall problem, revealing rapid evolution to a non-equilibrium state in infinite systems and a two-stage slow approach to equilibrium in finite systems, supported by simulations.

Contribution

It introduces a detailed analysis of the stochastic adiabatic wall, combining microscopic, thermodynamic, and numerical methods to understand its evolution and stationary states.

Findings

Infinite systems reach a non-equilibrium stationary state quickly.

Finite systems evolve in two stages: rapid then slow towards equilibrium.

Numerical simulations agree qualitatively with theoretical predictions.

Abstract

The time evolution of the adiabatic piston problem and the consequences of its stochastic motion are investigated. The model is a one dimensional piston of mass $M$ separating two ideal fluids made of point particles with mass $m\ll M$. For infinite systems it is shown that the piston evolves very rapidly toward a stationary nonequilibrium state with non zero average velocity even if the pressures are equal but the temperatures different on both sides of the piston. For finite system it is shown that the evolution takes place in two stages: first the system evolves rather rapidly and adiabatically toward a metastable state where the pressures are equal but the temperatures different; then the evolution proceeds extremely slowly toward the equilibrium state where both the pressures and the temperatures are equal. Numerical simulations of the model are presented. The results of the microscopical approach, the thermodynamical equations and the simulations are shown to be qualitatively in good agreement.