Dynamic Scaling, Coarsening and Stability in Physical Systems

物理系统中的动态缩放、粗化和稳定性

• 批准号: 0652558
• 负责人: Robert Pego
• 金额: $ 2.61万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652558&HistoricalAwards=false
• 关键词: Dynamic; Scaling; Coarsening; Stability; Physical

The main theme of the proposed work is on understanding the widespreadphenomenon of dynamic scaling in extended physical systems, buildingon stability analysis of coherent structures such as self-similarsolutions, nonlinear wave pulses and vortices. The paradox of dynamicscaling laws is that typically they can be predicted by back of theenvelope calculations, yet defy rigorous analysis for years.We will study scaling and stability in a variety of problems of highcurrent interest: the trend to self-similar behavior in Smoluchowski'smean-field model of coagulation; scaling and stability of fingeringpatterns in viscous fluid flows; models of domain coarsening in phasetransitions; existence and stability of solitary waves ininfinite-dimensional Hamiltonian systems; and stability of Bose-Einstein condensates with vortices. The work will involve the development of improved methods for stability analysis in nonlinear systems, for the rigorous analysis and numerical computation of self-similar solutions and nonlinear waves, and for physical modeling of step-terrace structures on crystalline surfaces.The general aim of this work is to improve understanding of large-scaledynamic behavior in complex systems. We aim to develop new and effective mathematical methods and concepts for the modeling and analysis of dynamic behavior in a variety of problems of high scientific interest. We study fundamental mathematical models for the large-scale dynamics of: agglomeration of small particles; mixing patterns in viscous fluid flow; microscopic structure of metallic alloys and semiconductor surfaces; waves on fluid interfaces; and quantum vortices in super-cold gases. Many issues that will be investigated are of fundamental interest in condensed-matter physics, aerosol physics, chemical engineering, and materials science.

这项工作的主题是基于自相似解、非线性波脉冲和涡流等相干结构的稳定性分析，了解扩展物理系统中动态标度的普遍现象。动力学标度定律的悖论在于，它们通常可以通过粗略的计算来预测，但多年来却无法进行严格的分析。我们将研究各种当前备受关注的问题中的标度和稳定性：Smoluchowski 的凝固平均场模型中的自相似行为的趋势；粘性流体流动中指法的缩放和稳定性；相变域粗化模型；无限维哈密顿系统中孤立波的存在性和稳定性；以及带有涡旋的玻色-爱因斯坦凝聚体的稳定性。这项工作将涉及开发非线性系统稳定性分析的改进方法，用于自相似解和非线性波的严格分析和数值计算，以及晶体表面阶梯结构的物理建模。这项工作的总体目标是提高对复杂系统中大规模动态行为的理解。 我们的目标是开发新的有效的数学方法和概念，用于对各种具有高度科学意义的问题的动态行为进行建模和分析。我们研究大规模动力学的基本数学模型：小颗粒的聚集；粘性流体流动的混合模式；金属合金和半导体表面的微观结构；流体界面上的波；以及超冷气体中的量子涡旋。 将要研究的许多问题对于凝聚态物理学、气溶胶物理学、化学工程和材料科学具有根本意义。