Mathematics of Collective Behavior: From Self-Organized Dynamics to Fluid Turbulence

集体行为数学：从自组织动力学到流体湍流

基本信息

批准号：
1813351
负责人：
Roman Shvydkoy
金额：
$ 28万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-15 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1813351&HistoricalAwards=false
关键词：
Mathematics Collective Behavior Self Organized

项目摘要

The theory of collective behavior is a rapidly developing area of mathematics that studies emergence of global phenomena in many biological, social and technological systems. Examples of such systems include flocks of birds, schools of fish, social networking, and exchange of political opinions among people. Although these systems arise from seemingly disconnected contexts, they share many common principles of self-organization. The project puts forward a comprehensive program of research to study those principles. More specifically, it aims to understand how "agents" in collective systems driven by their natural laws of communication can build global formations, such flocks, using only local interactions. This project will systematically classify types of global formations, their structure and stability under presence of an external force, such as gust of wind in the context of bird flocks or media influence in the context of opinion dynamics. On the way, the project will bring a connection between real applications and a new purely theoretical area of mathematics that studies spread of information in diffusive systems. The project will involve one graduate student, who will be involved in theoretical aspects as well as numerical and visual implementations of the proposed research. The technical implementation of the goals of the project involves modeling of emergent dynamics through a novel system of fractional parabolic equations. The system is designed to predict the end-state behavior of a "flock" through a careful long-time analysis of its global solutions. The new feature of proposed model involves inclusion of an adaptive anisotropic diffusion kernel as the main alignment mechanism of global behavior. This feature is not only mathematically motivated but is also observed in many biological and social congregations. The tools used in the analysis will be extended to study regularity problems in a more general class of parabolic and elliptic equations. The project will address validity of the classical laws such as energy conservation and will bring connection with the celebrated Onsager conjecture of 1949 that studies sharp regularity conditions under which such laws hold.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.

集体行为理论是一个快速发展的数学领域，研究许多生物、社会和技术系统中全球现象的出现。此类系统的例子包括鸟群、鱼群、社交网络以及人们之间的政治观点交换。尽管这些系统产生于看似互不相关的环境，但它们具有许多共同的自组织原则。该项目提出了一个全面的研究计划来研究这些原则。更具体地说，它的目的是了解集体系统中的“代理人”如何在自然沟通法则的驱动下，仅使用局部互动来构建全球形态，例如羊群。该项目将系统地对全球形态的类型、其结构和在外力存在下的稳定性进行分类，例如鸟群背景下的阵风或舆论动态背景下的媒体影响。在此过程中，该项目将在实际应用和新的纯理论数学领域之间建立联系，该领域研究扩散系统中的信息传播。该项目将涉及一名研究生，他将参与拟议研究的理论以及数值和视觉实施。该项目目标的技术实现涉及通过新颖的分数抛物线方程组对涌现动力学进行建模。该系统旨在通过对其全局解决方案进行仔细的长期分析来预测“羊群”的最终状态行为。所提出模型的新特征包括包含自适应各向异性扩散内核作为全局行为的主要对齐机制。这一特征不仅是出于数学原因，而且在许多生物和社会群体中也可以观察到。分析中使用的工具将扩展到研究更一般的抛物线和椭圆方程类中的规律性问题。该项目将解决能源守恒等经典定律的有效性问题，并将与 1949 年著名的 Onsager 猜想建立联系，该猜想研究此类定律成立的尖锐规律性条件。该奖项反映了 NSF 的法定使命，并通过评估被认为值得支持利用基金会的智力优势和更广泛的影响审查标准。