Stationary Motion of the Adiabatic Piston

TL;DR

This paper analyzes the stationary behavior of an adiabatic piston separating two ideal fluids with equal pressure but different temperatures, revealing that temperature differences induce piston motion despite pressure equality.

Contribution

It provides an explicit solution for the piston velocity distribution in the adiabatic piston problem, showing how temperature asymmetry causes macroscopic motion without net force.

Findings

Piston exhibits non-zero average velocity due to temperature difference.

Pressure remains equal on both sides despite piston motion.

Explicit velocity distribution derived up to second order in small parameter.

Abstract

We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressures but different temperatures T_1 and T_2, separated by an adiabatic movable piston whose mass M is much larger than the mass m of the fluid particules. This is the infinite version of the controversial adiabatic piston problem. The stationary non-equilibrium solution of the Boltzmann equation for the velocity distribution of the piston is expressed in powers of the small parameter \epsilon=\sqrt{m/M}, and explicitly given up to order \epsilon^2. In particular it implies that although the pressures are equal on both sides of the piston, the temperature difference induces a non-zero average velocity of the piston in the direction of the higher temperature region. It thus shows that the asymmetry of the fluctuations induces a macroscopic motion despite the absence of any macroscopic force. This same conclusion was previously obtained for the non-physical situation where M=m.