Mathematical Sciences: Large Scale Properties of Interacting Systems

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9504791 Quastel Abstract The investigator is studying the large scale behavior of interacting particle systems. In these models large systems of particles evolve statistically according to a simple local rule for which there are certain conserved quantities. The goal is to show that on a macroscopic scale the conserved quantities evolve according to deterministic conservation laws which usually take the form of nonlinear partial differential equations. Of particular interest are the transport coefficients and how they arise from the microscopic dynamics, in particular in the nongradient models where there is a net effect of fluctuations. Several types of nongradient models will be studied: Models of disordered systems, models of surfaces and interfaces, and lattice gas models for the incompressible Navier-Stokes equation. A fundamental, and still largely open, problem is to understand how large systems of microscopic particles organize themselves to produce the collective macroscopic behavior we see in the world around us. The investigator is pursuing this goal on models in which the microscopic dynamics has some degree of randomness. The research includes models of transport in disordered media which come from condensed matter physics, models of surfaces and interfaces, and lattice gas models for incompressible fluid flow. Besides providing a rigorous justification for the macroscopic conservation laws, other important consequences of these investigations are intrinsic formulas for transport coeffiaients, and precise estimates of the deviations of these microscopic systems from their macroscopic limits.