A finite-time exponent for random Ehrenfest gas

TL;DR

This paper introduces a finite-time exponent to characterize the dynamics of a random Ehrenfest gas, bridging the behavior between polygonal and circular scatterers and revealing how chaos emerges from pseudochaos.

Contribution

It proposes a finite-time exponent for the Ehrenfest gas and generalizes the reflection law to connect polygonal and circular scatterers, demonstrating the emergence of chaos.

Findings

Finite-time exponent characterizes Ehrenfest gas dynamics.

As polygons become circles, the exponent approaches the Lyapunov exponent.

Chaos emerges from pseudochaos in the limit of infinite sides.

Abstract

We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit.