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Free Path Lengths in Quasi Crystals

Journal article
Authors Bernt Wennberg
Published in Journal of Statistical Physics
Volume 147
Issue 5
Pages 981-990
ISSN 0022-4715
Publication year 2012
Published at Department of Mathematical Sciences, Mathematics
Pages 981-990
Language en
Keywords Lorentz gas, Quasi crystal, Free path lengths, periodic lorentz gas, boltzmann-grad limit, equation
Subject categories Mathematics


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 scatterers leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the free path length distribution asymptotically behaves as a power law. 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.

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