Free path lengths in quasi crystals

TL;DR

This paper investigates the distribution of free path lengths in a Lorentz gas model where scatterers are arranged as quasi crystals, revealing that such arrangements behave similarly to periodic crystals with non-exponential free path length distributions.

Contribution

It introduces the study of free path lengths in quasi crystal scatterer arrangements, extending understanding beyond periodic and random distributions.

Findings

Quasi crystals exhibit non-exponential free path length distributions.

Simulations show quasi crystals behave similarly to periodic crystals.

Distribution of free path lengths in quasi crystals differs from Poisson scatterers.

Abstract

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter $d$, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann-Grad limit, a Poisson distribution of scatters leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the distribution of free path behaves asymptotically like a Cauchy distribution. This paper considers the case when the scatters are distributed on a quasi crystal, i.e. non periodically, but with a long range order. Simulations of a one-dimensional model are presented, showing that the quasi crystal behaves very much like a periodic crystal, and in particular, the distribution of free path lengths is not exponential.