Mathematical Sciences: Interacting Particle Systems and Hydrodynamics

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9424270 Rezakhanlou Abstract An outstanding and long studied problem in statistical mechanics is to establish the connection between the microscopic structure of a fluid (or a gas) and its macroscopic behavior. Perhaps the most celebrated problem in this context is the derivation of the Hydrodynamic and kinetic equations from small scale dynamics governed by the Newton's second law. This problem is still wide open but some variants have been reaently treated. The investigator's research concerns the analysis of particle systems simulating a fluid or a dilute gas. Recently the investigator has derived the Boltzmann equation for the macroscopic particle densities for a class of one dimensional lattice gas models. Roughly speaking one shows that after a suitable scaling the microscopic particle density converges to a solution of the Boltzmann equation. The convergence is expected to be exponentially fast and the exponential rate can be expressed by a variational formula. 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 of mass, momentum and energy 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 and gases. 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.