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

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 M. For infinite systems it is shown that the piston evolves very rapidly toward a stationary non-equilibrium state with non-zero average velocity even if the pressures are equal but the temperatures different on both sides of the piston. For a 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.