The theory of equilibrium hyperbolic dynamical systems has been developed to the point where it now provides powerful mathematical tools to address open problems in physics. The principal investigator will focus on understanding the nature of Sinai-Ruelle-Bowen (SRB) measures and the statistics of relevant observations for nonequilibrium phenomena, using innovative approaches related to the study of chaotic systems. While sophisticated techniques have been developed and beautiful results obtained for equilibrium systems, the research topics of random and chaotic behavior in nonequilibrium systems have remained wide open. The reason for this is that chaotic phenomena in these settings have quite a different flavor from those in the equilibrium context. Recently developed mathematical tools in hyperbolic systems offer hope for significant progress. This project seeks to obtain both a theoretical understanding of and new ways to connect mathematical ideas to a variety of complex phenomena by addressing two challenging issues through the use of spectral analysis, the coupling method, and other innovative approaches. The first concerns statistical properties of nonequilibrium hyperbolic systems (e.g.,to prove the existence of SRB measures that characterize the steady states; to study the time correlation functions that relate to the diffusion matrix in the transport processes; to verify other limiting theorems for perturbed hyperbolic systems). Specific models include Lorentz gases under general forces, an ideal gas with slow-moving scatterers, and random systems with microstructure. The second issue has to do with properties of SRB measures and related physical laws (e.g., to understand the nature of SRB measures, including their entropy and Hausdorff dimensions; to obtain Ohm's law, the Einstein relation, and other physical laws for certain nonequilibrium hyperbolic systems that arise in physics).
This project will use mathematical tools to address applied problems in physics, chemical engineering, and other sciences. The theory of hyperbolic systems has provided excellent models or paradigms for understanding chaos and diffusion processes in systems that are random or changing over time. The goal is to capture the major complexity of these systems, from integrability to chaotic behavior, without the difficulty of integrating the equations of motion. As the study of physical invariant measures and their asymptotic statistical properties provides new insight into the nature of steady states and transport phenomena, the project will contribute to modern statistical physics and chemical engineering. Thus, it will have a broad impact outside of mathematics in the physical sciences and within the broader scientific community. The principal investigator plans to motivate and excite students about research on chaotic systems by directing graduate student research, by incorporating research-related concepts and simulations of nonequilibrium systems into new undergraduate courses, thereby enhancing research experiences for undergraduates (REUs), and by carrying out K-12 outreach activities. The research will be integrated into both graduate and undergraduate research projects, as well as into curriculum development. The principal investigator will seek institutional approval for a new topics course for graduate students on stochastic differential equations and a new undergraduate course that will focus on topics related to chaos and fractals. She will initiate a Guest Lecture Series at the Amherst Regional High School on topics related to chaos and fractal geometry, in order to engage local high school students in this interdisciplinary, cutting-edge research. She will also organize annual workshops for high school girls that will include hands-on scientific activities and career discussions with individual pursuing careers in the mathematical sciences, academic or otherwise. The project will broaden participation of women and underrepresented minorities and help them to visualize the beauty of mathematics.
