Free path lengths in quasicrystals

TL;DR

This paper studies the free path lengths in quasicrystalline scatterer configurations, proving the existence of a limit distribution that differs from random scatterer models, using advanced homogeneous space techniques.

Contribution

It establishes the first limit distribution for free path lengths in quasicrystals, extending kinetic transport theory beyond periodic and random models.

Findings

Limit distribution exists for quasicrystal scatterers.

Distribution differs from exponential distribution in random models.

Uses Ratner's measure classification for proofs.

Abstract

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g. at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of the Penrose tiling. Our main result proves the existence of a limit distribution of the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner's measure classification.