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

研究人员合作开发数学方法，以提高对复杂非线性系统动态行为的理解。重点是在各种物理环境中自相似或动态标度的普遍趋势，例如物理化学中胶体的聚类，流体力学和种群生物学模型中非线性波的选择，以及材料科学中纳米级岛屿的粗化。目标是根据动力学系统、偏微分方程和概率论的方法，设计一个统一的框架来处理这些问题。这项工作是由无限可分性的数学思想和概率论中重尾分布的极限定理之间的密切类比，以及这些不同物理环境中动态缩放和选择的明显不相关的问题所驱动的。这个项目的进展导致了对如何从复杂的各种问题中产生秩序的更深入的理解，这些问题在气溶胶物理学、流体力学、材料科学和物理化学等领域的基础科学兴趣。岛上粗化的工作有望改善纳米级台阶台阶结构在晶体表面的物理建模。关于凝结的工作旨在提高我们对云、烟、尘霾形成模型的理解，这对环境具有潜在的重要性。一个重要的成果是将看似不相关的问题的进展与一个基本的数学框架联系起来，该框架将概率论和动力系统理论联系在一起，并广泛地促进了见解从一个领域转移到另一个领域。

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.