RUI: Equilibrium and Nonequilibrium Dynamics for Systems of Physical Origin

In dynamical systems, systems that change with time such as planetary motion, much of the research is 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 systems that are influenced by other unknown systems, possibly on different scales.

This project will investigate properties of chaotic dynamical systems that are out of equilibrium, either due to the application of external forces or because mass or energy is allowed to escape. Systems to be studied during the course of this project include 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.

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 project will also support the involvement of undergraduate students in mathematics research. The highly visual nature and physical motivation of the problems outlined above will enable the principal investigator to recruit undergraduate students to participate in related research projects during the course of the project.

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 grant will contribute to the important goal of integrating research and education.

Motivated by the problems outlined above, this project is organized around 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 adapted Markov extensions (a generalization of Markov partitions). 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.