Scaling Limits for Microscopic Models

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A fundamental 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 models associated with the evolution of dilute gases, coagulation-fragmentation phenomenon and the formation of solids. As the first step, one derives a partial differential equation for the macroscopic evolution of such stochastic models. In the case of a dilute gas, one shows that after a suitable scaling, the particle density converges to a solution of the Boltzmann equation. In the case of the coagulation-fragmentation phenomenon, the rescaled microscopic particle density converges to a solution of the Smoluchowski equation. As a model of a crystal, one may consider an inhomogeneous Hamilton-Jacobi equation with viscosity. After a suitable scaling, the solutions converge to solutions of homogeneous Hamilton-Jacobi equation. Probabilistically, such a convergence is a law of large numbers and its corresponding central limit theorem and large deviation principle provide us with some vital information about the microscopic model under the study. 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. However these molecules manage to organize themselves in such a way as to form a flow pattern on a large scale. The investigator's research concerns the relationship between the microscopic structure and the macroscopic behavior of fluids, gases and solids. 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 fluids and gases. Moreover, interacting particle systems turn out to be the most efficient way of simulating the flow patterns of dilute gases.