Homogenization Methods in Statistical Physics

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In the physics and engineering of composite materials, it is natural to use models based on partial differential equations with highly oscillatory coefficients. The mathematical theory of homogenization was formed to answer the large class of analytic questions that arise from studying such equations. The general subject area has seen tremendous progress in the last ten years. However, despite the many past advances, some basic and fundamental phenomena still elude rigorous mathematical analysis. The principal investigator (PI) will develop new techniques and new perspectives for some of these remaining problems, with a focus on problems in statistical physics that can be viewed through the lens of homogenization. This includes models of semiconductors, fluid mixing, and droplet formation on rough surfaces. The project provides research training opportunities for graduate students. The PI will apply techniques from analysis and probability to study problems in statistical physics. The main focus will be on the large scale behavior of solutions of partial differential equations with highly oscillatory coefficients. This will include heat and wave equations with random coefficients as well as their lattice discretizations and related free boundary problems. Many of the proposed projects are inspired by Anderson localization phenomena. Another unifying theme is homogenization, where rough microscopic structure “averages” over large scales and gives rise to non-trivial smooth macroscopic effects. The primary goal is to obtain both qualitative and quantitative estimates for the large-scale behavior of solutions. The PI will utilize recent advances in homogenization and large scale regularity theory in order to attack a number of open problems. The primary areas of interest are Anderson localization, the Abelian sandpile, particle models for fluids, the mixing of fluids, and droplet formation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.