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中起着核心作用，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中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群集。这项建议的最终目标是开发新的分析和数值方法的动力学描述的复杂现象的基础上，多尺度和广泛的应用。目标是更好地理解当前研究的前沿问题，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，由来自马里兰州，布朗，爱荷华州，威斯康星州麦迪逊，亚利桑那州，奥斯汀-得克萨斯州和图卢兹，法国，谁将合并他们的专业知识，在建设，分析和实施动力学描述的一套选定的问题与交叉尺度从量子和微观尺度到宏观尺度的大学领导。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的例子，我们将使用动力学描述来研究复杂供应链的双曲流。这项研究计划的理论和建模方面，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“较小尺度的现象”。这一建议背后的理由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，动力学建模，动力学理论和相关的数值方法已经有了许多重大的发展，对物理和社会科学的新兴领域产生了相当大的影响。拟议的努力将大大加强美国研究人员在追求尖端研究和培养这一重要领域的新一代应用数学家方面可以发挥的主导作用。我们希望该项目通过对博士和博士后研究人员的高级培训，并通过提供与相关学科研究人员进行跨学科互动的平台，为科学劳动力的发展做出贡献。内部和外部的互动将通过协同合作，这将在三个年度研讨会期间将在马里兰州（第一年），法国和布朗（第二年），和威斯康星州（第三年）举行取得成果。国际会议将作为马里兰州大学科学计算和数学建模中心组织的一系列跨学科讲习班的一部分举行。项目研究人员将与CSCAMM的DOE多尺度等离子体动力学中心、爱荷华州州立大学的DOE艾姆斯实验室以及UT奥斯汀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.