CAREER: Anomalous Diffusion, Homogenization and Averaging

职业：异常扩散、均匀化和平均

• 批准号: 1252912
• 负责人: Gautam Iyer
• 金额: $ 48万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1252912&HistoricalAwards=false
• 关键词: CAREER; Anomalous; Diffusion; Homogenization; Averaging

This project studies various phenomena related to diffusive tracer particles in a strong array of opposing vorticies. One particularly interesting aspect is a seemingly anomalous diffusive behaviour exhibited on "intermediate" time scales. On long time scales this anomalous diffusive behaviour should "average" to an effectively Brownian behaviour, recovering classical homogenization results. The project crucially uses tools from both Probability and PDE. The educational component of this project aims to further interaction between these two areas from the undergraduate to the post-graduate level.Observing the movement of gravel particles shows intermittent periods of rest and travel. One aspect of this project studies an idealised system modelling this behaviour with the eventual aim of rigorously quantifying the "rest and travel". This is closely related to questions about homogenization -- the approximation of inhomogeneous media with microscopic inhomogeneities by a homogeneous one. This project crucially uses tools from two connected but different areas in Mathematics: Probability and PDE. The educational component of this project aims to further interaction between these two areas from the undergraduate level all the way to the post-graduate level.

本计画研究在强烈的反向涡阵中，与扩散示踪粒子相关的各种现象。一个特别有趣的方面是一个看似异常的扩散行为表现在“中间”的时间尺度。在长时间尺度上，这种异常的扩散行为应“平均”到一个有效的布朗行为，恢复经典的均匀化结果。该项目主要使用Probability和PDE工具。该项目的教育部分旨在促进这两个领域之间的互动，从本科到研究生水平。观察砾石颗粒的运动显示间歇性的休息和旅行。该项目的一个方面是研究一个理想化的系统，对这种行为进行建模，最终目的是严格量化“休息和旅行”。这与均匀化问题密切相关--用均匀介质近似具有微观不均匀性的不均匀介质。该项目关键地使用了数学中两个相互关联但不同的领域的工具：概率和PDE。该项目的教育部分旨在促进这两个领域之间的互动，从本科一直到研究生。