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Abstract: We investigate statistical properties of several classes of periodic billiardmodels which are diffusive. An introductory chapter gives motivation, and thena review of statistical properties of dynamical systems is given in chapter 2.In chapter 3, we study the geometry dependence of diffusion coefficients in atwo-parameter 2D periodic Lorentz gas model, including a discussion of how toestimate them from data. In chapter 4, we study the shape of position anddisplacement distributions, which occur in the central limit theorem. We showthat there is an oscillatory fine structure and what its origin is. This allowsus to conjecture a refinement of the central limit theorem in these systems. Anon-Maxwellian velocity distribution is shown to lead to a non-Gaussian limitdistribution. Chapter 5 treats polygonal billiard channels, developing apicture of when normal and anomalous diffusion occur, the latter being due toparallel scatterers in the billiard causing a channelling effect. We alsocharacterize the crossover from normal to anomalous diffusion. In chapter 6, weextend our methods to a 3D periodic Lorentz gas model, showing that normaldiffusion occurs under certain conditions. In particular, we construct anexplicit finite-horizon model, and we discuss the effect that holes inconfiguration space have on the diffusive properties of the system. We finishwith conclusions and directions for future research.

Autor: David P. Sanders Facultad de Ciencias, Universidad Nacional Autónoma de México


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