This research project is an analytical and numerical study of statistical transport under complex, non-conservative particle interactions. The models under study include kinetic integro-differential equations of Boltzmann type and semiclassical quantum probability transport models. Modeling, analysis, and development of numerical methods will be conducted in three primary problem areas: (1) Statistical models related to energy-dissipative phenomena, including hard sphere equations for kinetic models of rapid granular flow and dynamics of gas mixtures. (2) Mesoscale kinetic systems modeled by Boltzmann-Poisson and Vlasov-Poisson systems under strong shear forces induced by electric fields or by boundary driving forces. (3) Semiclassical quantum trajectory models for flow of quantum probability distributions, with application to problems in nanoscale modeling. New tools from non-linear analysis as well as new computational strategies will be developed to establish links from quantum to kinetic to fluid levels of modeling.
This research concerns the development and analysis of mathematical models for a number of important scientific problems. The models, which arise in biological, social, and physical sciences, describe statistically the transport of large numbers of interacting objects. Applications include modeling of rapid granular flows, chemically reacting gas mixtures, distribution of wealth in economic systems, opinion dynamics, and semiconductor behavior. This project will develop new mathematical tools that aid in the analysis of such models and their use in predicting the behavior of these complicated systems.
