Asymptotic properties of the Lorentz process and some closely related models

Abstract

Chaotic, “stochastic” behavior of deterministic systems is much interesting from both theoretical and applied points of view. An archetype of such systems is the Sinai billiard - or equivalently, its periodic extension, the periodic Lorentz process. The motivation for studying these models is multiple. In the physics literature, Hendrik Lorentz [L05] introduced Lorentz gas as a model of motion of electrons in a metal. By considering the dynamics of just one classical electron in a crystal, one obtains the (periodic) Lorentz process. Nowadays, a central problem in statistical physics is to derive macroscopic laws from microscopic dynamics. In the optimal case, the microscopic dynamics are Newtonian which makes the model more realistic. The two model families, where such rigorous results are available, are mathematical billiards and oscillators.

___Going back to the motivation by the work of Lorentz [L05], one sees that these kind of problems are also physically motivated (crystals often have impurities). In the last few years, some other nonhomogeneous modifications of the periodic Lorentz process were also considered, see for instance [SYZ12] for a very recent one. As both the delicate statistical properties of the periodic Lorentz process and the basic statistical properties of some non-homogeneous versions are current active research fields, there are plenty of interesting, challenging questions, a few of which we are going to address in this thesis.

___This thesis consists of six more or less self-contained chapters. Chapters 2 3 4 and 5 contain (almost verbatim) the articles [N11a, N11b, NSz12, NSzV12a], respectively. Chapter 6 is the preprint [NSzV12b], while Chapter 7 is an unpublished work, also joint with Domokos Szasz and Tamas Varju. I would also like to remark that Chapter 2 heavily overlaps with my MSc thesis. At several points - mainly in the introductions -, the Chapters may overlap (by not much, though). The high level logic of the thesis is the following: Chapters 2 and 3 are about some stochastic models (random walks) that are motivated by the periodic Lorentz processes. Chapter 4 is about a specific type of inhomogeneity (both in space and time) in Lorentz process. On the technical level, Chapters 2-4 require ideas almost exclusively from Probability theory. Chapter 5 suggests an approach to study general time inhomogeneity in dynamical systems (at its present state, not strong enough to treat two dimensional dynamics, though). Chapter 6 deals with Lorentz processes with infinite horizon in dimension d ≥ 3, while Chapter 7 is roughly speaking a new proof for the convergence to the Brownian motion in the plane, again, in the infinite horizon case. On the technical level, Chapters 5-7 require ideas primarily from the theory of Dynamical systems and elementary geometry, although Probability theory is still an important ingredient. We also mention that the motivation of Chapters 3 and 7 is mainly (but not exclusively) is the hope that they might be useful at attacking Conjecture 1.1. Each Chapter starts with an introductory Section and some of them has some remarks in the end pointing out some possible extensions and open questions. In the rest of this Section, we introduce each Chapter in some more details.