Coarse-graining, Renormalization, and Fractal Homogenization

This project is focused on the development of new mathematics for analyzing the statistical behavior of physical systems which exhibit complex behavior across a large number of length scales. A typical examples include turbulent fluids, such as the earth's atmosphere, which have fluctuations on the human scale (a gust of wind) and on the continental scale (weather patterns), and every scale in between. Other examples include important models in statistical mechanics and quantum field theory. Such chaotic physical systems have interesting behaviors which emerge through the interaction of these very different length scales, often called "critical phenomena" by physicists. Physicists have developed heuristic, non-rigorous ways of understanding and analyzing many such physical systems, some of which are called "renormalization group" arguments. One of the main goals of this project is to develop precise versions of these informal arguments which are mathematically rigorous. In the past decade, the work of the Principal Investigator (PI) and other mathematicians have led to a rigorous theory of "quantitative homogenization" of certain partial differential equations. These equations have some of the properties of the complex physical systems mentioned above, and the homogenization theory resembles renormalization group-type arguments in important ways. However, it currently works well only for problems with a small number of length scales. The project proposes to increase the level of sophistication of the homogenization methods until the theory can be deployed more flexibly on physical systems exhibiting critical behavior. This requires the development of new mathematical ideas and concepts and will require input from analysis, probability theory, partial differential equations and mathematical physics. The project provides research training opportunities for graduate students. The project has two main goals. The first one concerns improving the quantitative homogenization theory, so that it is more explicit in its dependence on important parameters in the equation (like the ellipticity ratio) and allows for degenerate and possibly unbounded coefficient fields. This is a well-known open problem in the subfield, but the PI and his collaborator Kuusi have made recent progress on this question, and this project will continue to develop these new ideas. A second focus of the project is to use these analytic methods developed for homogenization as means of formalizing heuristic renormalization group arguments in physics. Such methods arise in a wide variety of contexts, but the project has a few specific problems in mind. One arises in fluid turbulence, and concerns proving the anomalous diffusion of a passive scalar advected by a rough vector field. The PI and his collaborator Vicol have made recent progress on this question by using homogenization to formalize a renormalization group argument. This points the way to further possibilities, including the construction of more physically realistic examples of anomalous diffusion. Another potential application lies in Euclidean field theory, following a stochastic quantization approach to study Gibbs measures. 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.