CAREER: Anomalous Diffusion, Homogenization and Averaging

这个项目研究了与强大的相反涡旋阵列中的扩散示踪粒子有关的各种现象。一个特别有趣的方面是，在“中间”时间尺度上表现出一种看似异常的扩散行为。在长时间尺度上，这种反常的扩散行为应该“平均”到有效的布朗行为，恢复经典的均化结果。该项目关键地使用了概率和PDE的工具。该项目的教育部分旨在从本科生到研究生这两个领域之间的进一步互动。观察砾石颗粒的运动显示出间歇性的休息和旅行。该项目的一个方面是研究一种对这种行为进行模拟的理想化系统，最终目的是严格量化“休息和旅行”。这与均化问题密切相关--将具有微观不均质性的非均质介质近似为均质媒质。这个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：概率和偏微分方程。该项目的教育部分旨在促进这两个领域从本科生到研究生的互动。

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.