CAREER: Analysis of disordered systems

Michigan State University, East Lansing MI

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The project will focus on a rigorous analysis of wave propagation in disordered media, along with related analysis of random operators and matrices. The goal is to establish diffusive propagation of waves in a weakly disordered medium over arbitrarily long times. There is a rich non-rigorous theory of wave diffusion in the physics literature, based on heuristic analyses and uncontrolled perturbation theory, which suggests that diffusion occurs and predicts the leading order asymptotic behavior of the diffusion constant. In spite of the obvious importance of this problem, we are far from having a rigorous analysis of the mathematics involved. The PI will consider the problem from a number of view points, in particular using an ``augmented space" representation such as has proved useful in the analysis of random walks in random media and homogenization. Additionally the PI will consider related problems in the theory of random matrices, specifically random band matrices, with entries that vanish outside a band around the diagonal, which have been suggested as a model of a so-called metal-insulator transition as the bandwidth is moved from the matrix size down to one. What are the effects of disorder? This is a fundamental question in regards to any model of physics, even one in which disorder is not explicitly included. After all, any real world system is subject to a small amount of noise, and experience shows that even weak disorder may have a profound effect on the behavior of the system. Despite the fundamental nature of this subject, aspects of it remain poorly understood and it is not at present incorporated in the curriculum in an accessible way. This program seeks to bridge this gap as follows: 1) through the development of courses for undergraduates on the basic models of statistical mechanics and random matrix theory; and 2) by focused research on aspects of the question relevant to wave motion and semi-conductor physics. The goal of such work is to use analytical tools to further basic understanding of models of theoretical physics and applied mathematics. Such advances impact areas outside of mathematics and theoretical physics in the short run by yielding ideas for constructing new and better models useful in applications. In the long run, however, the impact will be even greater, as mathematical work illuminates those parts of physical theory that are truly fundamental.