FRG: Collaborative Research: Kinetic Description of Multiscale Phenomena: Modeling, Theory and Computation

动力学方程在数学物理的许多领域发挥着核心作用，从微观和纳米物理到连续介质力学。它们是物理和社会科学应用的数学描述中不可或缺的工具，从半导体、聚合物和等离子体到交通网络和蜂群。本建议的最终目标是发展基于多尺度复杂现象动力学描述的新型分析和数值方法，具有广泛的应用前景。目标是更好地理解当前研究前沿的问题，并通过理论、建模和数值的协同合作，为解决长期存在的问题做出贡献。为此，这个焦点研究小组（FRG）将提供一个平台，由来自马里兰大学、布朗大学、爱荷华州立大学、威斯康星-麦迪逊大学、亚利桑那州立大学、奥斯汀-德克萨斯大学和法国图卢兹大学的主要研究人员领导，他们将整合他们在构建、分析和实施动力学描述方面的专业知识，以解决一系列从量子和微观尺度到宏观尺度的交叉问题。讨论的主题包括微观和量子现象的动力学描述，以及宏观现象的动力学描述。作为动力学方法的一个新例子，我们将使用动力学描述来研究复杂供应链的双曲流。这个研究项目的理论和建模方面，在微观和宏观尺度上，将与基于动力学的数值方法相结合，以捕捉“更小尺度的现象”。这一建议背后的基本原理是及时努力解决现代应用数学中的几个重要问题。动力学理论并不新鲜。然而，在动力学建模、动力学理论和相关数值方法方面已经取得了许多重大进展，对物理科学和社会科学的新兴领域可能产生相当大的影响。这项提议的努力将大大加强美国研究人员在这一重要领域追求前沿研究和培养新一代应用数学家方面的主导作用。我们希望通过该项目为博士和博士后研究人员提供高级培训，并为相关学科的研究人员提供跨学科互动的平台，为科学劳动力的发展做出贡献。在马里兰州（一年级）、法国和布朗（二年级）以及威斯康星州（三年级）举行的三届年度研讨会期间，将通过协同合作保持内部和外部的互动。国际会议将作为由马里兰大学科学计算和数学建模中心（CSCAMM）组织的一系列跨学科研讨会的一部分举行。项目研究人员将与CSCAMM的DOE多尺度等离子体动力学中心，爱荷华州立大学的DOE Ames实验室以及UT Austin的计算工程与科学研究所（ICES）合作。

Kinetic equations play a central role in many areas of mathematical physics, from micro- and nano-physics to continuum mechanics. They are an indispensable tool in the mathematical description of applications in physical and social sciences, from semi-conductors, polymers and plasma to traffic networking and swarming. The ultimate goal of this proposal is to develop novel analytical and numerical methods based on kinetic descriptions of complex phenomena with multiple scales and with a wide range of applications. The objective is to achieve a better understanding of problems which are in the forefront of current research and to contribute to the solution of long-standing problems by synergetic collaboration of theory, modeling and numerics. To this end, this Focus Research Group (FRG) will provide a platform, led by leading researchers from Universities of Maryland, Brown, Iowa State, Wisconsin-Madison, Arizona State, Austin-Texas and Toulouse, France, who will merge their expertise in the construction, analysis and implementation of kinetic descriptions for a selected suite of problems with crossing scales from quantum and micro scales to the macro scales. Topics to be discussed include kinetic descriptions of microscopic and quantum phenomena, and kinetic descriptions of macroscopic phenomena. As a recent novel example for the kinetic methodology we will use kinetic descriptions to study hyperbolic flows for complex supply chains. The theoretical and modeling aspects of this research program, on both microscopic and macroscopic scales, will be integrated with kinetic-based numerical methods for capturing ``smaller scales phenomena". The rationale behind this proposal is a timely effort to address several important issues in modern applied mathematics. Kinetic theories are not new. Yet, there have been many major developments in kinetic modeling, kinetic theories and related numerical methods, with the potential for a considerable impact on emerging new fields in physical and social sciences. The proposed effort will significantly strengthen the leading role that the US researchers can play in pursuing cutting-edge research and training a new generation of applied mathematicians in this important field. We expect this project to contribute to the development of scientific workforce by advanced training for doctoral and postdoctoral researchers and by providing a platform for interdisciplinary interactions with researchers from related disciplines. Internal and external interactions will be maintained through synergetic collaborations which will come to fruition during the three annual workshops to be held in Maryland (Year 1), France and Brown (Year 2), and Wisconsin (Year 3). International meetings will be held as part of a series of interdisciplinary workshops organized by the Center for Scientific Computation and Mathematical Modeling (CSCAMM) at the University of Maryland. Project researchers will collaborate with the DOE Center for Multiscale Plasma Dynamics in CSCAMM, the DOE Ames Laboratory at Iowa State University, and the Institute for Computational Engineering and Sciences (ICES) at UT Austin.