Nonequilibrium Statistical Mechanics for Adiabatic Piston Problem

TL;DR

This paper develops a nonequilibrium statistical mechanics framework for the adiabatic piston problem, deriving heat transfer, steady velocity, and energy flux using local detailed balance and perturbation methods.

Contribution

It introduces a novel nonequilibrium approach to analyze the adiabatic piston, deriving explicit formulas for heat transfer and steady motion based on local detailed balance.

Findings

Derived heat transfer expressions between gases and the wall.

Obtained linear response formula for steady velocity and energy flux.

Calculated steady velocity up to order  in mass ratio .

Abstract

We consider the dynamics of a freely movable wall of mass $M$ with one degree of freedom that separates a long tube into two regions, each of which is filled with rarefied gas particles of mass $m$. The gases are initially prepared at equal pressure but different temperatures, and we assume that the pressure and temperature of gas particles before colliding with the wall are kept constant over time in each region. We elucidate the energetics of the setup on the basis of the local detailed balance condition, and then derive the expression for the heat transferred from each gas to the wall. Furthermore, by using the condition, we obtain the linear response formula for the steady velocity of the wall and steady energy flux through the wall. Using perturbation expansion in a small parameter $\epsilon\equiv\sqrt{m/M}$, we calculate the steady velocity up to order $\epsilon$.