Efficient algorithms for general periodic Lorentz gases in two and three dimensions

TL;DR

This paper introduces efficient algorithms for simulating particle trajectories in periodic Lorentz gases in two and three dimensions, especially effective in the Boltzmann-Grad limit, with applications to calculating free path length distributions.

Contribution

The paper develops novel algorithms that improve efficiency for Lorentz gas trajectory calculations in 2D and 3D, applicable to general crystal lattices and near the Boltzmann-Grad limit.

Findings

Algorithms efficiently compute trajectories as obstacle radius approaches zero.

Application to distribution of free path lengths near the Boltzmann-Grad limit.

Extension of algorithms to general crystal lattices.

Abstract

We present efficient algorithms to calculate trajectories for periodic Lorentz gases consisting of square lattices of circular obstacles in two dimensions, and simple cubic lattices of spheres in three dimensions; these become increasingly efficient as the radius of the obstacles tends to 0, the so-called Boltzmann-Grad limit. The 2D algorithm applies continued fractions to obtain the exact disc with which a particle will collide at each step, instead of using periodic boundary conditions as in the classical algorithm. The 3D version incorporates the 2D algorithm by projecting to the three coordinate planes. As an application, we calculate distributions of free path lengths close to the Boltzmann-Grad limit for certain Lorentz gases. We also show how the algorithms may be applied to deal with general crystal lattices.