Nonlocal Variational Problems from Physical and Biological Models

物理和生物模型的非局部变分问题

• 批准号: 2306962
• 负责人: Ihsan Topaloglu
• 金额: $ 20.21万
• 依托单位: Virginia Commonwealth University
• 依托单位国家: 美国
• 项目类别: Standard 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=2306962&HistoricalAwards=false
• 关键词: Nonlocal; Variational; Problems; Physical; Biological

Nonlocal models are used to describe a wide array of physical phenomena in material science, quantum mechanics, mathematical physics, and biology. The pertinence of these models is that they introduce scales used to investigate microstructures in macroscopic domains, as in the case of the nonlocal variational models characterizing the collective behavior in biological systems. This project involves the mathematical analysis of nonlocal attractive-repulsive interaction energies that are directly connected to aggregations models for biological and robotic swarming, granular media, and self-assembly of nanoparticles. The emphasis is on understanding the competing effects of interactions on the physical systems considered. This will yield insight into the general phenomenology of nonlocality and ultimately provide predictions of collective behavior in these systems and guide the design of improved devices. Training of undergraduate and graduate students will be integrated in the research project, which will lead to the discovery of new results through student research projects. The project has three main mathematical aims: (1) Develop new tools to study symmetry of optimizers of a model describing the distribution of oppositely charged phases; (2) study the effect of regularization via an interfacial free energy in swarming models described by attractive-repulsive nonlocal energies; and, finally, (3) introduce an extension of Gamow’s liquid drop model to investigate the effect of neutrons in determining the shape of the nucleus of an atom and the threshold of nuclear fission. To pursue these goals, techniques varying from nonlinear to geometric analysis, optimization and partial differential equations will be implemented. These techniques will be combined with computer-based experimentation. The common aim of all three directions is understanding structures of optimizers in a variety of physical systems of practical interest.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.

非局域模型用于描述材料科学、量子力学、数学物理学和生物学中的各种物理现象，这些模型的相关性在于它们引入了用于研究宏观领域中的微观结构的尺度，就像非局域的情况一样。该项目涉及对非局域吸引-排斥相互作用能量的数学分析，这些能量与生物和机器人集群、颗粒介质和自组装的聚合模型直接相关。重点是了解相互作用对所考虑的物理系统的竞争影响，这将深入了解非局域性的一般现象学，并最终提供对这些系统中集体行为的预测，并指导改进设备的设计。研究生将融入该研究项目，这将通过学生研究项目发现新结果。该项目具有三个主要数学目标：（1）开发新工具来研究描述分布的模型的优化器的对称性。带相反电荷的相； (2) 研究通过吸引-排斥非局域能量描述的蜂群模型中的界面自由能进行正则化的效果；最后，(3) 引入伽莫夫液滴模型的扩展来研究中子在确定形状中的影响；为了实现这些目标，将从非线性到几何分析、优化和偏微分方程等技术与基于计算机的实验相结合。所有三个方向的目标都是了解各种具有实际意义的物理系统中优化器的结构。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。