  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.
