Collective Behavior of Stochastic Systems

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The investigator's research concerns stochastic and deterministic models associated with the evolution of gases, coagulation and fragmentation of clusters and the formation of crystals. As the first step, one derives a partial differential equation for the macroscopic evolution of such stochastic models. Roughly, after a suitable scaling, the density of particles in a gas with grazing collisions converges to a solution of Landau Equation, the density of coagulating and fragmenting clusters converges to a solution of Smoluchowski Equation, and a rough interface modeled by Hamilton-Jacobi PDE with inhomogeneity (impurity) homogenizes to a homogeneous Hamilton-Jacobi equation. Many issues related to these models are not understood. Large-deviations type questions for the homogenization and gases would provide us valuable informations about the corresponding microscopic models. The phenomenon of gelation for the Smoluchowski Equation is still a mystery and our understanding of the corresponding microscopic phenomenon should help resolving some open problems in this context. Our world appears differently at different scales! For example a fluid or a gas is a collection of an enormous number of molecules that collide incessantly and move erratically without any particular aim. How do these molecules then manage to organize themselves in such a way as to form a flow pattern on a large scale? Roughly the reason is that the local conservation laws impose constraints not immediately visible on the microscopic scale. The investigator's research concerns the relationship between the microscopic structure and the macroscopic behavior of fluids, gases and crystals. The analysis of the mathematical models consisting of a large number of interacting particles is proved to be useful in understanding the intricate behavior of our microscopic world such as the formation of gels, the roughness of the surface boundary of solids, the occurrence of sharp corners in crystals,....