Dynamics of a Massive Piston in an Ideal Gas: Oscillatory Motion and Approach to Equilibrium

TL;DR

This paper investigates the dynamic behavior of a massive piston in an ideal gas, revealing oscillatory motion and thermalization, with detailed scaling laws for instability onset and relaxation time.

Contribution

It provides a combined numerical and heuristic theoretical analysis of piston dynamics, highlighting the instability onset and approach to equilibrium in a large system.

Findings

Piston exhibits damped oscillations after initial quiescence.

System thermalizes to a Maxwellian velocity distribution.

Instability onset time scales as L log L.

Abstract

We study numerically and theoretically (on a heuristic level) the time evolution of a gas confined to a cube of size $L^3$ divided into two parts by a piston with mass $M_L \sim L^2$ which can only move in the $x$-direction. Starting with a uniform ``double-peaked'' (non Maxwellian) distribution of the gas and a stationary piston, we find that (a) after an initial quiescent period the system becomes unstable and the piston performs a damped oscillatory motion, and (b) there is a thermalization of the system leading to a Maxwellian distribution of the gas velocities. The time of the onset of the instability appears to grow like $L \log L$ while the relaxation time to the Maxwellian grows like $L^{7/2}$.