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.

非局部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微观结构的量表，例如在表征生物系统中集体行为的非局部变异模型的情况下。该项目涉及对非本地吸引力 - 抑制性相互作用能量的数学分析，这些相互作用能与生物学和机器人摆动，颗粒状培养基以及纳米颗粒自组装的聚集模型直接连接。重点是了解相互作用对所考虑的物理系统的竞争影响。这将洞悉非局部性的一般现象，并最终提供这些系统中集体行为的预测，并指导改进的设备的设计。本科生和研究生的培训将集成到研究项目中，这将通过学生研究项目发现新的结果。该项目具有三个主要的数学目的：（1）开发新的工具来研究描述替代充电阶段分布的模型的优化器的对称性； （2）研究通过有吸引力的驱动性非局部能量描述的蜂群模型中调节的效果；最后，（3）引入了Gamow液滴模型的扩展，以研究确定原子核的形状和核裂变阈值的中子的作用。为了实现这些目标，将实施从非线性到几何分析，优化和部分微分方程的技术不同。这些技术将与基于计算机的实验结合使用。这三个方向的普遍目的是了解优化者在各种实际关注的物理体系中的结构。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准来通过评估来诚实地获得支持。