Collaborative Research: Scaling and infinite divisibility in models of coarsening and other dynamic selection problems

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

Menon 0605006 Pego 0604420 The investigators collaborate in developing mathematical methods that improve the understanding of dynamic behavior in complex nonlinear systems. The focus is on the universal trend to self-similarity, or dynamic scaling, in a variety of physical settings, such as clustering of colloids in physical chemistry, selection of nonlinear waves in fluid mechanics and models of population biology, and coarsening of nanoscale islands in materials science. The goal is to devise a unified framework for the treatment of these problems based on methods from dynamical systems, partial differential equations, and probability theory. The work is driven by close analogies between the mathematical ideas of infinite divisibilty and limit theorems for heavy-tailed distributions in probability theory, and apparently unrelated problems of dynamic scaling and selection in these diverse physical settings. Progress in this program leads to deeper understanding of how order emerges from complexity in a variety of problems of fundamental scientific interest in the areas of aerosol physics, fluid mechanics, materials science, and physical chemistry. The work on island coarsening promises to improve the physical modeling of nanoscale step-terrace structures on crystalline surfaces. The work on coagulation aims to improve our understanding of models for the formation of clouds, smoke, dust and haze, which is of potential environmental importance. A significant outcome is to link progress on apparently unconnected problems with a basic mathematical framework that ties together probability and dynamical systems theory and that broadly facilitates the transfer of insights from one area to another.