# Mesoscopic description of the adiabatic piston: kinetic equations and $\mathcal H$-theorem

## TL;DR

This paper develops a mesoscopic kinetic theory for the adiabatic piston problem, deriving equations that describe the system's evolution towards equilibrium and proving an H-theorem for entropy increase.

## Contribution

It introduces a novel kinetic description of the adiabatic piston problem, including a generalized molecular chaos assumption and proof of the H-theorem.

## Key findings

- Equilibrium corresponds to a steady-state solution of the kinetic equations.
- The Boltzmann entropy with piston motion satisfies the H-theorem.
- Results extend to systems with short-range repulsive potentials.

## Abstract

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.