Mesoscopic description of the adiabatic piston: kinetic equations and $\mathcal H$-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 from the Liouville equation, 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 $\mathcal H$-theorem. The results are generalized to any short-ranged repulsive potentials among particles and include the ideal gas as a limiting case.