Scaling limits of random and deterministic diffusion processes on the integer lattice

整数晶格上随机和确定性扩散过程的尺度限制

• 批准号: 1606670
• 负责人: Charles Smart
• 金额: $ 10.21万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1606670&HistoricalAwards=false
• 关键词: Scaling; limits; random; deterministic; diffusion

The principal investigator will study mathematical models of microscopic interactions in solid materials. More specifically, the PI will study the scaling limits of such models, which capture macroscopic effects of microscopic structure. The PI is interested in three important classes of models arising from statistical physics. First, there are cluster growth models for corrosion of metals and crystal formation. Second, there are models of electrical properties of randomly composited materials. Third, there are models which lie in the intersection of the previous two categories, such as those for fluid flow in porous media. Many unresolved questions surround these models and are now of great interest in probability theory. The PI aims to resolve some of these questions by (1) using computer simulation to identify new phenomena and (2) applying tools from analysis, combinatorics, and probability to rigorously understand the new phenomena.The models alluded to above share a common theme; each is a diffusion processes on the integer lattice whose scaling limit can be interpreted as the solution of a nonlinear elliptic partial differential equation (PDE). The PI is primarily interested in three models: the Abelian sandpile, random walks in random environments, and first passage percolation. The PI will study the interplay between the asymptotic statistics of the diffusion process and the regularity properties of the solutions of the limiting PDE. Since the two sides here are tightly coupled, it is often possible and fruitful to transfer results between them. The strategy will be to carefully adapt the regularity machinery of the limiting PDE to the discrete setting, since this occasionally leads to the discovery of new phenomena.

主要研究者将研究固体材料中微观相互作用的数学模型。 更具体地说，PI将研究这些模型的标度极限，这些模型捕捉微观结构的宏观效应。 PI对统计物理学中产生的三种重要模型感兴趣。 首先，有金属腐蚀和晶体形成的簇生长模型。 第二，有随机复合材料的电性能模型。 第三，存在位于前两个类别的交叉点的模型，例如多孔介质中的流体流动的模型。 许多悬而未决的问题围绕着这些模型，现在在概率论中引起了极大的兴趣。 PI的目标是通过以下方式解决其中一些问题：（1）使用计算机模拟来识别新现象;（2）应用分析、组合学和概率学工具来严格理解新现象。上面提到的模型有一个共同的主题;每个模型都是整数网格上的扩散过程，其缩放极限可以解释为非线性椭圆偏微分方程（PDE）的解。 PI主要对三个模型感兴趣：阿贝尔沙堆，随机环境中的随机行走和首次通过渗流。 PI将研究扩散过程的渐近统计与极限PDE解的正则性之间的相互作用。 由于这里的双方是紧密耦合的，因此在他们之间转移成果往往是可能的和富有成效的。 该策略将是仔细调整的规则性机制的限制PDE的离散设置，因为这偶尔会导致发现新的现象。