Scaling Dynamics of a Massive Piston in a Cube Filled With Ideal Gas: Exact Results

TL;DR

This paper provides exact results and proofs for the scaling dynamics of a massive piston in a large cube filled with ideal gas, confirming heuristic differential equations for the system's evolution.

Contribution

It offers rigorous proofs and exact results for the previously heuristic scaling behavior of a piston-gas system in a large container.

Findings

Confirmed the autonomous coupled differential equations for piston and gas distribution.

Provided exact mathematical results supporting the scaling regime.

Validated previous heuristic arguments with rigorous proofs.

Abstract

We continue the study of the time evolution of a system consisting of a piston in a cubical container of large size $L$ filled with an ideal gas. The piston has mass $M\sim L^2$ and undergoes elastic collisions with $N\sim L^3$ gas particles of mass $m$. In a previous paper, Lebowitz, Piasecki and Sinai considered a scaling regime, with time and space scaled by $L$, in which they argued heuristically that the motion of the piston and the one particle distribution of the gas satisfy autonomous coupled differential equations. Here we state exact results and sketch proofs for this behavior.