Deterministic Diffusion in Periodic Billiard Models

TL;DR

This paper explores the statistical and geometric properties of periodic billiard models, revealing how diffusion varies with system parameters, and refining the understanding of distribution shapes and diffusion types in these dynamical systems.

Contribution

It provides new insights into the geometry dependence of diffusion, distribution shapes, and the conditions for normal and anomalous diffusion in periodic billiard models, including extensions to 3D systems.

Findings

Diffusion coefficients depend on system geometry.

Position and displacement distributions exhibit oscillatory fine structures.

Normal diffusion occurs under specific conditions in 3D models.

Abstract

We investigate statistical properties of several classes of periodic billiard models which are diffusive. An introductory chapter gives motivation, and then a review of statistical properties of dynamical systems is given in chapter 2. In chapter 3, we study the geometry dependence of diffusion coefficients in a two-parameter 2D periodic Lorentz gas model, including a discussion of how to estimate them from data. In chapter 4, we study the shape of position and displacement distributions, which occur in the central limit theorem. We show that there is an oscillatory fine structure and what its origin is. This allows us to conjecture a refinement of the central limit theorem in these systems. A non-Maxwellian velocity distribution is shown to lead to a non-Gaussian limit distribution. Chapter 5 treats polygonal billiard channels, developing a picture of when normal and anomalous diffusion occur, the latter being due to parallel scatterers in the billiard causing a channelling effect. We also characterize the crossover from normal to anomalous diffusion. In chapter 6, we extend our methods to a 3D periodic Lorentz gas model, showing that normal diffusion occurs under certain conditions. In particular, we construct an explicit finite-horizon model, and we discuss the effect that holes in configuration space have on the diffusive properties of the system. We finish with conclusions and directions for future research.