Collaborative Research: Scaling and Infinite Divisibility in Models of Coarsening and Other Dynamic Selection Problems

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Menon0605006Pego0604420 The investigators collaborate in developing mathematicalmethods that improve the understanding of dynamic behavior incomplex nonlinear systems. The focus is on the universal trendto self-similarity, or dynamic scaling, in a variety of physicalsettings, such as clustering of colloids in physical chemistry,selection of nonlinear waves in fluid mechanics and models ofpopulation biology, and coarsening of nanoscale islands inmaterials science. The goal is to devise a unified framework forthe treatment of these problems based on methods from dynamicalsystems, partial differential equations, and probability theory. The work is driven by close analogies between the mathematicalideas of infinite divisibilty and limit theorems for heavy-taileddistributions in probability theory, and apparently unrelatedproblems of dynamic scaling and selection in these diversephysical settings. Progress in this program leads to deeper understanding ofhow order emerges from complexity in a variety of problems offundamental scientific interest in the areas of aerosol physics,fluid mechanics, materials science, and physical chemistry. Thework on island coarsening promises to improve the physicalmodeling of nanoscale step-terrace structures on crystallinesurfaces. The work on coagulation aims to improve ourunderstanding of models for the formation of clouds, smoke, dustand haze, which is of potential environmental importance. Asignificant outcome is to link progress on apparently unconnectedproblems with a basic mathematical framework that ties togetherprobability and dynamical systems theory and that broadlyfacilitates the transfer of insights from one area to another.