Dynamics of a Massive Piston in an Ideal Gas

TL;DR

This paper analyzes the behavior of a massive piston in an ideal gas, deriving hydrodynamical equations that describe the system's dynamics in the large container limit, and demonstrating convergence of the piston’s motion to these equations.

Contribution

It introduces a scaling regime where the piston’s dynamics converge to autonomous hydrodynamical equations in the large system limit.

Findings

Piston motion converges to hydrodynamical equations as system size grows.

Derived autonomous coupled equations for piston and gas distribution.

Discussed long-term heuristic dynamics of the system.

Abstract

We study a dynamical system consisting of a massive 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 $N\sim L^3$ non-interacting gas particles of mass $m=1$. We find that, under suitable initial conditions, there is, in the limit $L \to \infty$, a scaling regime with time and space scaled by $L$, in which the motion of the piston and the one particle distribution of the gas satisfy autonomous coupled equations (hydrodynamical equations), so that the mechanical trajectory of the piston converges, in probability, to the solution of the hydrodynamical equations for a certain period of time. We also discuss heuristically the dynamics of the system on longer intervals of time.