Statistical Properties of Hyperbolic Systems with Singularities

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is devoted to the rigorous investigation of statistical properties of dynamical systems and their applications to the physical sciences. The primary goal is to study hyperbolic systems with singularities. Most of those under study in the project fall under the chaotic billiards heading, which includes popular physical models such as the Lorentz and hard-ball gases that provide the mathematical foundations of statistical mechanics. These systems have properties similar to those of geodesic flows on negatively curved manifolds and Axiom-A diffeomorphisms and flows, but their singularities give rise to an unpleasant fragmentation of phase space, which makes them much harder to study. In particular, dynamical systems with weak statistical properties (such as slow decay of correlations) play extremely important roles in applications to physical sciences. The dynamics in such systems are intermittent between regular and chaotic. However, mathematical methods for the analysis of systems with slow mixing rates were developed only recently and are still difficult to apply to realistic models. Based upon recent results of Young, Dolgopyat, and Chernov, the principal investigator proposes to develop new approaches to estimating the decay of correlations that can be applied to more general systems with singularities, systems that could not be handled by existing techniques.The first major component of this project deals with systems with singularities that represent the most general mathematical models of physical phenomena. Of special interest in the project is the subject of "chaotic billiards," a term used by mathematicians and physicists to describe certain phenomena that are observed in the study of Boltzmann's ergodic hypothesis. The second key component of the project focuses on ergodic theory, which is an important mathematical research field in which the subjects of probability and dynamical systems come together. Through a collaboration with Feres, the principal investigator will explore the scattering properties of gas-surface collisions through random billiards (again the mathematical concept, not the parlor game). This demonstrates the contribution of the rigorous study of abstract singular systems to the concrete field of chemical engineering. Indeed, such a study may ultimately lead to new techniques for gas separation in industrial processes. It is expected that the progress in understanding statistical properties of dynamical systems that will be made in this project will have an impact on our knowledge of many important events in nature and find application in other scientific and engineering disciplines.