Invariance Principle for the Random Wind-Tree Process

TL;DR

This paper proves that a particle moving through a random array of cubes exhibits diffusive behavior, converging to Brownian motion under specific scaling limits, extending previous results to a new geometric setting.

Contribution

The authors adapt a coupling method to establish an invariance principle for the wind-tree model, a new geometric configuration, building on prior work with spherical scatterers.

Findings

Convergence to Brownian motion in the diffusive limit

Extension of coupling method to cube scatterers

Validation of invariance principle for wind-tree model

Abstract

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes - the random wind-tree model introduced in Ehrenfest-Ehrenfest (1912). We show that, in the joint Boltzmann-Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to a Brownian motion in a particular scaling limit. In a previous paper (2019) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, the same strategy with some modification can be used to prove an invariance principle for the random wind-tree model.