9704565 Rezakhanlou 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 recently treated. The investigator's research concerns the analysis of particle systems modeling fluids and gas. Recently the investigator in collaboration with James Tarver has derived a Boltzmann type equation for the macroscopic particle densities of a class stochastic particle systems modeling dilute gases. Roughly speaking one shows that after a suitable scaling the microscopic particle density converges to a solution of a Boltzmann type equation. The investigator also shows that the convergence is exponentially fast and derives an explicit expression for the exponential rate of convergence. This result allows us to go beyond the macroscopic description given by the Boltzmann equation and obtain some valuable insight about the microscopic structure of the particle system under 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. 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 has 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.
