RUI: Statistical Properties of Nonequilibrium and Extended Dynamical Systems

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The research funded by this grant will focus in large part on the study of mathematical models of particle systems with collision interactions that are central to the field of statistical mechanics, and in particular to our understanding of chaotic dynamical systems. Such systems play an important role in the study of non-equilibrium dynamics, which model, for example, the motion of particles subjected to external forces or nonelastic collisions. Other examples of non-equilbrium dynamics include systems in which mass or energy is allowed to escape, and large-scale systems of smaller interacting components which exchange mass or energy. Such systems have been used in theoretical physics and chemical engineering to model atom traps, explore mechanisms for heat conduction in solids and investigate metastability in molecular processes. Rigorous mathematical results obtained during the course of this grant will aid in the interpretation of these studies as well as suggest new directions of inquiry. This grant also supports the involvement of undergraduates in mathematics research. Using the highly visual nature and physical motivation of the problems outlined above, the author will recruit undergraduate students to work on these topics during each summer funded by the grant. Special emphasis will be given to recruiting students from underrepresented groups in research mathematics. Students will disseminate results of their research at poster sessions and through publication in undergraduate or research journals, as appropriate. By stimulating interest in research careers in mathematics and creating a peer community supportive of that interest, the grant will contribute to the important goal of integrating research and education.Much research in dynamical systems focuses on closed systems in which the dynamics are self-contained. In many modeling situations, however, such a global view is not possible and it becomes necessary to study local systems that are influenced by other unknown systems, possibly on different scales. Such considerations motivate many of the systems to be studied during the course of this grant: systems in which mass or energy may enter or exit through deterministic or random mechanisms. The grant is organized around three specific projects: The first project investigates the statistical properties of both classical and non-equilibrium particle systems, which constitute an important class of models from statistical mechanics as described above; the second concerns open systems, which are inspired by physical models in which mass or energy is allowed to escape; the third project investigates the behavior of dynamical systems which are comprised of (possibly infinitely many) smaller components linked together, with orbits or energy allowed to pass between them. When focused on one component at a time, such systems generalize the discussion of open systems in a natural way by allowing both entry and escape. The intellectual merit of the research activity funded by the grant stems from the depth of the analytical tools to be developed as well as the complexity and variety of the systems under consideration. Using his recent work concerning the spectral decomposition of the transfer operator for dispersing particle systems, the author will investigate both equilibrium and non-equilibrium models from statistical mechanics. This approach is expected to resolve a longstanding conjecture by Bowen and Ruelle regarding the continuous time flow and to provide rigorous analysis of physically important quantities such as entropy production. A second tool the author will use is the construction of Markov extensions (a generalization of finite and countable Markov partitions), which make no Markovian assumptions on the dynamics and are widely applicable to nonuniformly hyperbolic systems, including Hénon maps and a wide variety of particle systems. The application of such tools to systems out of equilibrium, open coupled map lattices and extended systems will represent significant advances in the study of such systems. Efforts to understand these tools in one context strengthens them and aids in their application to other areas of mathematics. Their intellectual interest is enhanced by the application of these ideas to resolve problems posed and approached formally in the physics literature.