Normal diffusion in crystal structures and higher-dimensional billiard models with gaps

TL;DR

This paper demonstrates that three-dimensional periodic Lorentz gases with certain gap sizes exhibit normal diffusion, supported by heuristic and numerical evidence, and extends these findings to higher-dimensional billiard models including hard-sphere fluids.

Contribution

It provides new insights into diffusion behavior in 3D and higher-dimensional billiard models with gaps, showing conditions under which normal diffusion occurs despite infinite horizons.

Findings

Normal diffusion occurs in 3D Lorentz gases with sufficiently small gaps.

Numerical simulations confirm the diffusive regimes in the models.

Results extend to higher-dimensional billiard systems, including fluids.

Abstract

We show, both heuristically and numerically, that three-dimensional periodic Lorentz gases -- clouds of particles scattering off crystalline arrays of hard spheres -- often exhibit normal diffusion, even when there are gaps through which particles can travel without ever colliding, i.e., when the system has an infinite horizon. This is the case provided that these gaps are not "too big", as measured by their dimension. The results are illustrated with simulations of a simple three-dimensional model having different types of diffusive regime, and are then extended to higher-dimensional billiard models, which include hard-sphere fluids.