The Kompaneets Equation, Anomalous Diffusion, and Brownian Entanglement

该研究项目致力于对三种自然现象的调查中出现的问题进行数学研究；这些研究通过使用相关的数学工具而结合起来。第一组问题涉及从数学上理解光子在低密度等离子体中的行为。第二组问题涉及流体流动中粒子的行为。在某些重要的应用场合，粒子以不同寻常的方式扩散。这个项目在中间（相对于长）时间尺度上研究这种“异常”扩散行为。第三组问题的动机是用数学方法量化长聚合物分子的纠缠；它旨在量化相关随机游走轨迹的缠绕。每一个研究项目都涉及研究生和博士后，他们将在工作过程中得到广泛的培训。本研究将使用应用分析、偏微分方程和概率论的工具。第一组问题涉及Kompaneets方程的长期行为，并侧重于理解玻色-爱因斯坦凝聚体的形成。第二组问题研究了流体环境下被动标量输运的小噪声极限和其他相关模型。虽然这些方程的长期行为是众所周知的，但这里的兴趣是在小噪声的中间时间区域，在那里观察到异常扩散效应，并且极限行为是由时间变化的布朗运动和时间分数方程描述的。第三组问题研究具有边界的紧致黎曼流形上布朗轨迹的长时间“缠绕”或“纠缠”。这与覆盖空间的热核估计密切相关，并受到长聚合物分子缠结研究的推动。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project is devoted to mathematical study of questions arising in the investigation of three natural phenomena; these studies are united by the use of related mathematical tools. The first set of questions concerns mathematically understanding behavior of photons in low-density plasmas. The second set of questions concerns the behavior of particles in fluid flow. In certain situations important for applications, particles diffuse in unusual ways. This project investigates such "anomalous" diffusive behavior on intermediate (as opposed to long) time scales. The third set of questions is motivated by mathematically quantifying the entanglement of long polymer molecules; it aims to quantify the winding of trajectories of the related random walks. Each of these research projects involves graduate students and postdoctoral associates, who will be broadly trained through the course of their work. This research will use tools from applied analysis, partial differential equations, and probability. The first set of questions concerns the long-time behavior of the Kompaneets equation, and focuses on understanding the formation of a Bose-Einstein condensate. The second set of questions studies the small-noise limit of passive scalar transport and other related models arising in the context of fluids. While the long-time behavior of these equations is well known, here interest is in the small-noise intermediate-time regime where an anomalous diffusive effect is observed, and the limiting behavior is described by a time changed Brownian motion and time-fractional equations. The third set of questions studies the long-time "winding" or "entanglement" of Brownian trajectories on compact Riemannian manifolds with boundary. This is closely related to heat kernel estimates on covering spaces, and is motivated by the study of the entanglement of long polymer molecules.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.