Distribution of the linear flow length in a honeycomb in the small-scatterer limit

TL;DR

This paper analyzes the distribution of free path lengths in a honeycomb billiard with small scatterers, proving the existence of a limiting distribution and explicitly calculating it for randomly initiated trajectories.

Contribution

It provides the first explicit computation of the free path length distribution in a honeycomb billiard with small scatterers, extending understanding of billiard dynamics in such geometries.

Findings

Existence of a limiting distribution of free path lengths.

Explicit formula for the distribution in the small-scatterer limit.

Applicable to random initial conditions in honeycomb billiards.

Abstract

We study the statistics of the linear flow in a punctured honeycomb lattice, or equivalently the free motion of a particle on a regular hexagonal billiard table with holes of equal size at the corners and obeying the customary reflection rules. In the small-scatterer limit we prove the existence of the limiting distribution of the free path length with randomly chosen origin of the trajectory and explicitly compute it.