Large scale phenomena in models of statistical mechanics

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

The project addresses a variety of mathematical problems at the borderline of probability theory and statistical mechanics. The common feature of these is a relatively simple formulation, intriguing underlying phenomena and deep connections to physical disciplines. The first round of problems pertains to the Random Conductance Model. These include derivation of scaling limits of random walks in disordered media, study of spectral properties of random Laplacians, analysis of fluctuations of effective conductivity, detailed control of heat-kernel decay, etc. The second area proposed to investigate is that of Gradient Fields. These problems are ubiquitous in physical and applied sciences but their mathematical understanding is currently stalled because of the inability to overcome non-convexity of the underlying interaction. We offer specific methods and approaches how this obstacle may technically be overcome. The third class of problems addresses the area of Disordered Systems. The specific subjects of interest are random-field spin models and the dynamics governed by the parabolic Anderson model. Precise mathematical approaches are outlined that could lead to significant improvements in the mathematical understanding of these problems. A good many of listed problems are devised with the intention to provide training, and inclusion in research, of graduate students and postdocs who have interest in Probability Theory and Mathematical Physics. The tremendous success of hard sciences in accurate description, modeling and even forecasting complex natural phenomena derives, in large part, from their foundation in rigorous mathematics. In the context of solid-state physics and material sciences, the mathematical methods most commonly used are those of probability and/or statistics. This comes as no surprise as these disciplines have been designed precisely to deal with systems involving large numbers of individual constituents. The present project studies three specific classes of problems in probability theory of large systems whose origin is rooted in physics of materials. The most pertinent general question we ask is how the structural details of materials, and their various inherent irregularities, exactly express themselves in their macroscopic properties. As a rule, we seek an enveloping principle, or a "physical law", that extracts, in quantitative or qualitative form, the essential features from the specifics. It is hoped that developing the theoretical foundations of such systems will eventually lead to significant advances in engineering applications. There is also an immediate impact for mathematics itself: analysis of complex phenomena inevitably involves a number of separate sub-disciplines of mathematics and will thus lead to fruitful exchange of ideas among them. The project naturally offers an opportunity to include graduate students and postdocs into the research environment at a top research US university.