On the Second Law of thermodynamics and the piston problem

TL;DR

This paper analytically demonstrates that a microscopic model of the piston problem, with specific initial conditions and assumptions, obeys the Second Law of thermodynamics, linking microscopic dynamics to macroscopic thermodynamic laws.

Contribution

It provides an explicit analytical derivation showing the piston problem model obeys the Second Law of thermodynamics under certain conditions.

Findings

The model's evolution aligns with the Second Law.

The piston remains in equilibrium without drift.

Analytical solutions from Liouville equation confirm thermodynamic consistency.

Abstract

The piston problem is investigated in the case where the length of the cylinder is infinite (on both sides) and the ratio $m/M$ is a very small parameter, where $m$ is the mass of one particle of the gaz and $M$ is the mass of the piston. Introducing initial conditions such that the stochastic motion of the piston remains in the average at the origin (no drift), it is shown that the time evolution of the fluids, analytically derived from Liouville equation, agrees with the Second Law of thermodynamics. We thus have a non equilibrium microscopical model whose evolution can be explicitly shown to obey the two laws of thermodynamics.