On the distribution of free-path lengths for the periodic Lorentz gas III

Abstract

In a flat 2-torus with a disk of diameter $r$ removed, let ${\Phi }_r(t)$ be the distribution of free-path lengths (the probability that a segment of length larger than $t$ with uniformly distributed origin and direction does not meet the disk). We prove that ${\Phi }_r(t/r)$ behaves like $\frac{2}{{\pi }^{2}t}$ for each $t>2$ and in the limit as $r\to 0^{+}$, in some appropriate sense. We then discuss the implications of this result in the context of kinetic theory.

26 pages, 5 figures, to be published in Commun. Math. Phys

References in corpus (1)

• The statistics of the trajectory in a certain billiard in a flat two-torus

Cited by in corpus (10)

• On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas
• On the Periodic Lorentz Gas and the Lorentz Kinetic Equation
• The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions
• Free path lengths in quasicrystals
• Recent Results on the Periodic Lorentz Gas
• Power-law distributions for the free path length in Lorentz gases
• Free path lengths in quasi crystals
• Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes
• Efficient algorithms for general periodic Lorentz gases in two and three dimensions
• Convergence to the equilibrium for the kinetic transport equation in the two-dimensional periodic Lorentz Gas