Scaling Limits via Stochastic Homogenization

通过随机均质化缩放限制

• 批准号: 1712841
• 负责人: Charles Smart
• 金额: $ 21万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712841&HistoricalAwards=false
• 关键词: Scaling; Limits; via; Stochastic; Homogenization

This research project concerns the statistical properties of mathematical models known as particle systems that describe fluid-dynamical and materials-science aspects of important physical systems. More specifically, the project studies models of (1) cluster growth, (2) randomly composited materials, and (3) diffuse gases. Many basic and important questions about these models remain open. One of the more striking examples is that we still lack a fully rigorous understanding how fluid equations arise from molecular interactions. The investigator will attempt to make progress on this problem and others by applying recently-developed techniques in the theory of stochastic homogenization. The work will have an interdisciplinary character, mixing results from probability and analysis. The project will make use of computer simulation to empirically test ideas during development of the theory.Techniques in quantitative stochastic homogenization, particularly those used to obtain optimal rates in the elliptic setting, are now mature enough to attack problems that previously did not seem to be amenable to this approach. The basic homogenization idea remains the same, namely, to view the microscopic models as perturbations of their smooth macroscopic limits. The key new insight is that the regularity theory of the macroscopic limit often has a microscopic analogue. Examples of possible applications include the derivation of the Landau equation as a scaling limit of interacting particles and Anderson localization for a Bernoulli potential on the lattice. This approach has already proved successful, and the investigator expects it to continue to yield new results.

这项研究项目涉及被称为粒子系统的数学模型的统计特性，这些模型描述了重要物理系统的流体动力学和材料科学方面。更具体地说，该项目研究了(1)团簇生长、(2)随机合成材料和(3)扩散气体的模型。关于这些模型的许多基本而重要的问题仍然悬而未决。一个更引人注目的例子是，我们仍然缺乏对流体方程是如何从分子相互作用中产生的完全严格的理解。研究人员将试图通过应用随机齐化理论中最新发展的技术来在这个问题和其他问题上取得进展。这项工作将具有跨学科的特点，混合了概率和分析的结果。该项目将利用计算机模拟在理论发展过程中对想法进行经验性测试。定量随机均化技术，特别是那些用于在椭圆环境中获得最优比率的技术，现在已经足够成熟，可以解决以前似乎不适合这种方法的问题。基本的均匀化思想保持不变，即将微观模型视为其光滑宏观极限的扰动。关键的新见解是，宏观极限的正则性理论往往有微观上的类比。可能应用的例子包括作为相互作用粒子的标度极限的朗道方程的推导和晶格上伯努利势的安德森局部化。这一方法已经被证明是成功的，研究人员预计它将继续产生新的结果。

项目成果

A Free Boundary Problem with Facets

带面的自由边界问题

• DOI: 10.1007/s00205-018-1323-4
• 发表时间: 2019
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Feldman, William M.;Smart, Charles K.
• 通讯作者: Smart, Charles K.