RUI: Nonuniformly Hyperbolic Dynamical Systems out of Equilibrium

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This award funds research dedicated to the study of chaotic dynamical systems out of equilibrium. 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. These considerations motivate many of the systems to be studied during the course of this award: systems in which mass or energy may enter or exit through deterministic or random mechanisms, and large-scale systems of smaller interacting components that 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 award will aid in the interpretation of these studies as well as suggest new directions of inquiry. This award also supports the involvement of undergraduates in mathematics research. Using the highly visual nature and physical motivation of the problems outlined above, the PI will recruit undergraduate students to work on these topics during each summer funded by the award. Special emphasis will be given to recruiting students from underrepresented groups in research mathematics. Students will disseminate results of their research via poster sessions, conference presentations and publications in peer-reviewed journals. By stimulating interest in research careers in mathematics and creating a peer community supportive of that interest, the award will contribute to the important goal of integrating research and education.Motivated by the problems outlined above, this award is organized around three specific projects: The first project investigates the statistical properties of both classical and non-equilibrium particle systems with collision interactions, an important class of models from statistical mechanics; the second concerns open systems, which relate on the one hand to models of physical systems in which mass or energy is allowed to escape, and on the other to the study of metastable states; the third project concerns the behavior of extended dynamical systems comprised of a finite or infinite number of smaller components linked together, with orbits or energy allowed to pass between them. Important examples include the aperiodic Lorentz gas and mechanical models of heat conduction. Using his recent work concerning spectral decompositions of transfer operators for dispersing particle systems, the PI will investigate equilibrium and non-equilibrium models from statistical mechanics and provide rigorous analysis of physically important quantities such as entropy production. In addition, the PI will adapt and develop techniques using projective cones due to Birkhoff as well as the construction of Markov extensions (a generalization of Markov partitions) to the study of such systems. None of these techniques require Markovian assumptions on the dynamics, making them widely applicable to a wide variety of nonuniformly hyperbolic and physically important systems. The application of these tools to central models from equilibrium and non-equilibrium statistical mechanics 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.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.