RUI: Nonuniformly Hyperbolic and Extended Dynamical Systems

Fairfield University, Fairfield CT

Investigators

Abstract

The PI will investigate the properties of chaotic dynamical systems that are out of equilibrium due to the influence of either external forces or interconnected components. Research in dynamical systems is often focused 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 influenced by external dynamics, possibly on different spatial or temporal scales. To better understand these phenomena, the PI will study open systems in which mass or energy may enter or exit through deterministic or random mechanisms, as well as large-scale systems of smaller interacting components that exchange mass or energy. These problems are strongly motivated by connections with statistical mechanics and seek to advance our understanding of fundamental questions related to energy transport and diffusion. This award will also support the involvement of undergraduates in mathematics research. The highly visual nature and physical motivation of the problems will enable the investigator to recruit undergraduate students to participate in related research projects. 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, this award will contribute to the important goal of integrating research and education. The funded research includes three specific projects. The first project investigates the statistical and thermodynamic 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 physical systems in which mass or energy is allowed to escape, and on the other to the study of metastable states. The third project generalizes open systems to include linked and extended dynamical systems comprised of two or more components that exchange mass or energy through deterministic or random mechanisms. Important examples include the aperiodic Lorentz gas and mechanical models of heat conduction. The investigator will bring to bear several analytical techniques that he has been instrumental in developing for these classes of systems, including his recent work concerning the spectral decomposition of transfer operators for dispersing particle systems, contractions in projective cones due to Birkhoff, and the construction of Markov extensions adapted to open 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 techniques 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.