Mesoscopic Description of the Adiabatic Piston: Kinetic Equations and $H$-Theorem

The adiabatic piston problem is solved at the mesoscale using a kinetic theory approach. The problem is to determine the evolution towards equilibrium of two gases separated by a wall with only one degree of freedom (the adiabatic piston). A closed system of equations for the distribution functions of the gases conditioned to a position of the piston and the distribution function of the piston is derived, under the assumption of a generalized molecular chaos. It is shown that the resulting kinetic description has the canonical equilibrium as a steady-state solution. Moreover, the Boltzmann entropy, which includes the motion of the piston, verifies the $H$-theorem. The kinetic description is not limited to the thermodynamic limit nor to a small ratio between the masses of the particle and the piston, and collisions among particles are explicitly considered.