Scaling Limits via Stochastic Homogenization

该研究项目涉及被称为粒子系统的数学模型的统计特性，该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)团簇生长模型，(2)随机复合材料模型，(3)扩散气体模型。关于这些模型的许多基本和重要的问题仍然没有解决。一个更显著的例子是，我们仍然缺乏对流体方程如何从分子相互作用中产生的完全严谨的理解。研究者将尝试通过应用随机均匀化理论中最新发展的技术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思想进行实证检验。定量随机均匀化技术，特别是那些用于在椭圆环境中获得最佳速率的技术，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。基本的均质化思想保持不变，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常常有微观的类似物。可能应用的例子包括作为相互作用粒子尺度极限的朗道方程的推导和晶格上伯努利势的安德森局域化。这种方法已经被证明是成功的，研究者希望它能继续产生新的结果。

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.