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.

集体行为的理论是数学的快速发展领域，研究了许多生物，社会和技术系统中全球现象的出现。这种系统的例子包括鸟类，鱼类学校，社交网络以及人们之间的政治意见交流。尽管这些系统源于看似断开的环境，但它们具有许多共同的自我组织原则。该项目提出了一项全面的研究计划，以研究这些原则。更具体地说，它旨在了解其自然沟通定律驱动的集体系统中的“代理”如何仅使用局部互动来建立全球形成，即羊群。该项目将系统地分类全球形成的类型，其在外部力量存在下的结构和稳定性，例如在鸟羊群的背景下阵风或在意见动力学的背景下媒体的影响。在途中，该项目将在实际应用程序与新的数学纯粹理论领域之间建立联系，从而在扩散系统中扩展信息。该项目将涉及一名研究生，他们将参与理论方面以及拟议研究的数值和视觉实现。该项目目标的技术实施涉及通过新型的分数抛物线方程系统对新兴动态进行建模。该系统旨在通过对其全球解决方案进行仔细的长期分析来预测“羊群”的最终行为。提出的模型的新特征涉及将自适应各向异性扩散核作为全球行为的主要对准机制。此功能不仅是出于数学动机，而且在许多生物学和社会众中也被观察到。分析中使用的工具将扩展以研究一类更一般的抛物线和椭圆方程中的规律性问题。该项目将解决经典法律的有效性，例如节能，并将与1949年著名的Onsager猜想联系，该猜想研究了此类法律所持有的尖锐规律性条件。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来通过评估来支持的。