Many interesting problems in science and engineering involve systems with large numbers of interacting constituents. Typical examples are gases and fluids, which are made out of many atoms, as well as granular media like sand. An important question for such systems is how to describe their behavior at large scales. For example, under what conditions is the collective dynamics of a large number of grains of sand similar to the behavior of a fluid, and when is it not? Questions like this are of considerable interest from a purely scientific point of view as well as for industrial applications ranging from coal processing to pharmaceutical and grain/food preparation lines. The objective of this research project is to advance the mathematical understanding of how microscopic dynamics gives rise to macroscopic behavior. Necessarily this requires study of specialized models. In this project several such models suitable for rigorous mathematical analysis are under study.
Very popular mechanical models in the mathematical studies of classical statistical mechanics are collisions between scatters, in particular dispersing billiards. Recent advances have shown that such systems have very strong statistical properties. However, these results are limited to a very small number of degrees of freedom. The PI proposes to rigorously investigate statistical properties of specific models having many degrees of freedom. Since the rigorous analysis of purely mechanical models with many degrees of freedom is mathematically very challenging, many models in statistical physics introduce additional random elements. This project will also mathematically study the hydrodynamic limit of a special class of randomly interacting particle systems, which are in fact very closely related to the models under study. For this purpose the PI will extend the techniques that were recently developed to analyze of the hydrodynamic limit of the dissipative Boltzmann equation.
