Rigorous Results for the Periodic Oscillation of an Adiabatic Piston

TL;DR

This paper rigorously analyzes the periodic oscillation of an adiabatic piston, demonstrating convergence of its motion to an averaged behavior using advanced averaging techniques, and extends results to multiple pistons and smoothed interactions.

Contribution

It provides the first rigorous proof of the piston’s averaged behavior in a multi-dimensional setting, including convergence rates and uniformity over initial conditions and smoothing.

Findings

Convergence of piston motion to averaged behavior on M^{1/2} time scale

Rate of convergence is O(M^{-1/2}) for 1D gas particles

Results extend to multiple pistons and smoothed interactions

Abstract

We study a heavy piston of mass $M$ that moves in one dimension. The piston separates two gas chambers, each of which contains finitely many ideal, unit mass gas particles moving in $d$ dimensions, where $ d\geq 1$. Using averaging techniques, we prove that the actual motions of the piston converge in probability to the predicted averaged behavior on the time scale $M^ {1/2} $ when $M$ tends to infinity while the total energy of the system is bounded and the number of gas particles is fixed. Neishtadt and Sinai previously pointed out that an averaging theorem due to Anosov should extend to this situation. When $ d=1$, the gas particles move in just one dimension, and we prove that the rate of convergence of the actual motions of the piston to its averaged behavior is $\mathcal{O} (M^ {-1/2}) $ on the time scale $M^ {1/2} $. The convergence is uniform over all initial conditions in a compact set. We also investigate the piston system when the particle interactions have been smoothed. The convergence to the averaged behavior again takes place uniformly, both over initial conditions and over the amount of smoothing. In addition, we prove generalizations of our results to $N$ pistons separating $N+1$ gas chambers. We also provide a general discussion of averaging theory and the proofs of a number of previously known averaging results. In particular, we include a new proof of Anosov's averaging theorem for smooth systems that is primarily due to Dolgopyat.