HYPERBOLIC DYNAMICS IN PHYSICAL MODELS

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The project is devoted to the study of mathematical models in statistical mechanics. It deals with hyperbolic dynamical systems that have their origins in physical studies of electrical current, conductivity, diffusion, and gases of molecules. Hyperbolicity is a key feature used to derive laws of electrodynamics and thermodynamics, or otherwise to describe the macroscopic behavior of the underlying systems of particles. Specific models to be treated in the project include gases of hard balls (in particular, those emulating Brownian motion), Lorentz gases (imitating electrons in metals), and the Galton board (a physics analogue of a pinball machine). The proposal also deals with the dynamics of fluids governed by the Navier-Stokes equations and quasi-geostrophic equations. The main tools of the proposed studies are methods of hyperbolic dynamics, averaging theories, and stochastic processes. The proposal deals with problems originating in physics, electrodynamics, thermodynamics, and statistical mechanics. Its main goal is to study complex physical phenomena by exact mathematics and derive precise laws of nature. In many cases the principal investigator hopes to obtain rigorous mathematical descriptions of physical processes that have been only known empirically or qualitatively. In some cases the mathematical analysis reveals new features of complex evolution models that have been overlooked or unnoticed in experimental studies. The principal investigator closely collaborates with physicists and statisticians on several join research projects.