The Simplest Piston Problem I: Elastic Collisions

TL;DR

This paper investigates the complex dynamics of three elastic particles in a finite interval, revealing oscillatory and chaotic behaviors, and predicts a power-law distribution for piston crossing times using a billiard mapping.

Contribution

It introduces a novel billiard mapping approach to analyze the piston problem and predicts a power-law tail for crossing time distributions, verified by simulations.

Findings

Piston exhibits oscillatory and chaotic motion.

Distribution of crossing times follows a -3/2 power-law tail.

Billiard mapping effectively explains complex piston dynamics.

Abstract

We study the dynamics of three elastic particles in a finite interval where two light particles are separated by a heavy ``piston''. The piston undergoes surprisingly complex motion that is oscillatory at short time scales but seemingly chaotic at longer scales. The piston also makes long-duration excursions close to the ends of the interval that stem from the breakdown of energy equipartition. Many of these dynamical features can be understood by mapping the motion of three particles on the line onto the trajectory of an elastic billiard in a highly skewed tetrahedral region. We exploit this picture to construct a qualitative random walk argument that predicts a power-law tail, with exponent -3/2, for the distribution of time intervals between successive piston crossings of the interval midpoint. These predictions are verified by numerical simulations.