On some random billiards in a tube with superdiffusion

TL;DR

This paper studies a class of random billiards in a tube with stochastic reflection angles, deriving a non-standard Central Limit Theorem for particle displacement that highlights superdiffusive behavior and the absence of a second moment.

Contribution

It introduces a new probabilistic model of random billiards with stochastic reflections and establishes a non-standard CLT for the horizontal displacement.

Findings

Derived a non-standard CLT for particle displacement

Identified superdiffusive behavior in the model

Showed the second moment does not exist under the invariant measure

Abstract

We consider a class of random billiards in a tube, where reflection angles at collisions with the boundary of the tube are random variables rather than deterministic (and elastic) quantities. We obtain a (non-standard) Central Limit Theorem for the horizontal displacement of a particle, which marginally fails to have a second moment w.r.t.\ the invariant measure of the random billiard.