Anomalous Diffusion: Physical Origins and Mathematical Analysis

反常扩散：物理起源和数学分析

• 批准号: 2306254
• 负责人: Liping Liu
• 金额: $ 26.42万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306254&HistoricalAwards=false
• 关键词: Anomalous; Diffusion; Physical; Origins; Mathematical

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.

扩散是颗粒或分子的物理过程逐渐均匀地分布在环境介质中，例如，墨水在水中的散布。异常扩散是指该过程显着偏离其常规行为的实例，并且以两种不同的方式表现出来：当溶质颗粒移动的速度较慢时，溶质颗粒覆盖较短的时间标准的较大距离时，超扩散的速度较高，表现出更不稳定的行为。 扩散决定了复杂系统（例如人体）中材料（例如电导率或热传导率）和现象的重要物理特性。通过研究异常扩散的物理起源和数学复杂性，该项目将增强对不同领域扩散过程的基本理解，从物理学和化学到生物学和环境科学。科学的目标是建立粒子的微观结构和物理特性与环境介质与可测量的广义扩散率及其缩放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因素会影响扩散过程，例如各向异性，异质性和颗粒的变形性，以及培养基和外部应用电磁场的粘弹性，并旨在从微观差分方程中扩展到微观差异分析的微观差分方程。目的是扩展经典的Stokes-Einstein框架，并应对与电力法，非标准布朗动作和异常扩散有关的技术挑战。在微观层面上，该项目考虑了管理粒子行为和环境介质的完整集合场方程。在宏观水平上，目的是在不同时间尺度下得出颗粒的浓度或概率分布函数的有效进化方程。这涉及在不同时间制度内的运动显微镜方程和识别不同的扩散行为。使用多尺度分析方法来揭示分数微分方程的数学和物理起源。通过探索这些方程式如何出现以及它们的解决方案如何影响异常扩散，该项目将对潜在的物理机制有了更深入的了解。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和广泛影响的评估来评估值得通过评估来支持的。

项目成果

Polynomial inclusions: Definitions, applications, and open problems

• DOI: 10.1016/j.jmps.2023.105440
• 发表时间: 2023-09
• 期刊: Journal of the Mechanics and Physics of Solids
• 影响因子: 5.3
• 作者: Tianyu Yuan;Liping Liu