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.
The project has been devoted to mathematical models of naturalphenomena and practical tasks in various sciences. Its main focus ison statistical mechanics (studies of gases and fluids). Itsadditional areas include image processing, computer vision, packingand cutting problems, and optimization of statistical and numericalalgorithms. All these studies have clear applied character andaffect a broad range of human activities, such as medical research,industry, finance, software development, etc. The main goal of the project is the creation and enhancement ofadequate mathematical tools for the respective practical tasks. Inparticular, the proposal deals with Brownian motion, diffusion,electrical current and superconductivity, chaotic motion in apinball machine (Galton board), as well as general dynamics of gasesand fluids governed by Navier-Stokes equations and quasi-geostrophicequations. The project involves theoretical studies and computerexperimentation. A notable byproduct of our work is the developmentof more efficient computer algorithms and programs for practicalapplications. The results of the project are published in books,journal articles, disseminated at conferences and schools.Electronic products (databases, code) are posted on various webpages. The Principal investigator and his co-authors have publishedtwo books, about 20 journal articles, and delivered several dozentalks at professional meetings. The principal investigator trainedseveral Ph.D. students and undergraduate students. After graduationthose students continue productive research on their own and findemployment at leading American universities and abroad, includingUniversity of Massachusetts and Warwick University (UK).
