Anomalous Diffusion: Physical Origins and Mathematical Analysis

扩散是粒子或分子在环境介质中逐渐均匀分布的物理过程，如墨滴在水中的扩散。异常扩散是指过程明显偏离其通常行为的情况，它以两种不同的方式表现出来：亚扩散是指溶质颗粒比预期移动得慢得多，超扩散是指溶质颗粒在更短的时间尺度内覆盖更大的距离，表现出更不稳定的行为。扩散决定了材料的重要物理特性（如导电性或导热性）和复杂系统（如人体）中的现象。通过研究异常扩散的物理起源和数学复杂性，该项目将加强对扩散过程在不同领域的基本理解，从物理和化学到生物和环境科学。科学目标是建立粒子和周围介质的微观结构和物理性质与可测量的广义扩散率及其标度定律之间的定量关系。该项目将为研究生提供研究培训机会。研究者将考虑观察到的影响扩散过程的因素，如颗粒的各向异性、非均质性和可变形性，以及介质和外源电磁场的粘弹性，并利用系统的多尺度分析方法，从微观随机微分方程扩展到宏观分数阶微分方程。目标是扩展经典的斯托克斯-爱因斯坦框架，并解决与幂定律、非标准布朗运动和异常扩散相关的技术挑战。在微观层面上，该项目考虑了控制粒子和周围介质行为的多物理场方程的完整集合。在宏观层面上，目的是推导出粒子在不同时间尺度上的浓度或概率分布函数的有效演化方程。这包括粗粒化微观运动方程和识别不同时间范围内的不同扩散行为。采用多尺度分析方法揭示分数阶微分方程的数学和物理根源。通过探索这些方程是如何出现的，以及它们的解是如何影响异常扩散的，该项目将对潜在的物理机制有更深入的了解。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。

Diffusion is the physical process of particles or molecules gradually become uniformly distributed in an ambient medium, e.g., the dispersal of an ink drop in water. Anomalous diffusion refers to instances when the process deviates significantly from its usual behavior, and it manifests in two distinct ways: subdiffusion when solute particles move much slower than expected, and superdiffusion when solute particles cover larger distances in shorter timescales, exhibiting more erratic behavior. Diffusion dictates important physical properties of materials (e.g., electrical or heat conductivities) and phenomena in complex systems (e.g., human body). By studying the physical origins and mathematical intricacies of anomalous diffusions, the project will enhance the fundamental understanding of diffusion processes in diverse fields, ranging from physics and chemistry to biology and environmental science. The scientific goal is to establish the quantitative relationship between the microstructural and physical properties of the particle and ambient medium and the measurable generalized diffusivity and its scaling laws. The project will provide research training opportunities for graduate students. The investigator will consider factors that are observed to affect diffusion processes, such as anisotropy, heterogeneity, and deformability of particles, and the viscoelasticity of the medium and externally applied electromagnetic fields and aim to scale up from microscopic stochastic differential equations to macroscopic fractional differential equations, using a systematic multiscale analysis approach. The goal is to extend the classical Stokes-Einstein framework and address technical challenges related to power laws, non-standard Brownian motions, and anomalous diffusions. On a microscopic level, the project considers the complete set of multiphysics field equations governing the behavior of particles and the ambient medium. On a macroscopic level, the aim is to derive effective evolution equations for the concentration or probability distribution function of particles at different time scales. This involves coarse-graining the microscopic equation of motion and identifying distinct diffusional behaviors within different time regimes. Multiscale analysis methods are employed to uncover the mathematical and physical origins of fractional differential equations. By exploring how these equations emerge and how their solutions impact anomalous diffusions, the project will result in a deeper understanding of the underlying physical mechanism.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.