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。重点是各种物理环境中自相似或动态缩放的普遍趋势，例如物理化学中的胶体聚集，流体力学和种群生物学模型中的非线性波的选择，以及材料科学中纳米尺度岛的粗化。其目标是根据动力学系统、偏微分方程组和概率论的方法，设计出一个处理这些问题的统一框架。这项工作是由概率论中关于重尾分布的无限可除性和极限定理的数学思想与这些不同物理环境下的动态缩放和选择显然无关的问题之间的密切相似推动的。这一计划的进展使我们更深入地理解了在气溶胶物理、流体力学、材料科学和物理化学领域中的基本科学兴趣是如何从各种问题的复杂性中产生秩序的。岛状粗化的工作有望改进晶体表面纳米级阶梯结构的物理模拟。凝聚方面的工作旨在提高我们对云、烟雾、尘埃和雾霾形成模型的理解，这对环境具有潜在的重要意义。一个重要的结果是，将在明显不相关的问题上取得的进展与一个基本的数学框架联系在一起，该框架将概率和动力系统理论联系在一起，并广泛地促进将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.