An outstanding and long studied problem in statistical mechanics is to establish the connection between the microscopic world and its macroscopic behavior. The investigator's research concerns stochastic and deterministic models associated with the evolution of gases, the formation of gels and crystals, and Hamiltonian systems with random potentials. 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, a rough interface modeled by Hamilton-Jacobi PDE with impurity homogenizes to a homogeneous Hamilton-Jacobi equation, and a chain of particles with random potential are governed by an effective potential. Many issues related to these models are not fully understood. Large-deviations type questions for the homogenization and a central limit theorem type result for gases would provide us with valuable information about the corresponding microscopic models. Also, simplified models for coagulation phenomenon should help us to discover the rules governing the mysterious interaction between gels and sols.

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. As an another example, consider a solid to which further material sticks from the ambient atmosphere. The process of the attachment is a function of a huge variety of growth mechanisms depending on the materials involved, their temperature, composition, etc. Following the tradition of statistical mechanics, one studies simplified models which nevertheless captures some of the essential physics.

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 components is proved to be useful in understanding the intricate behavior of our microscopic world such as the formation of crystals, the emergence of clots in blood, the roughness of the surface boundary of solids and occurrence of shocks in fluids.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tomek Bartoszynski
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University of California Berkeley
United States
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