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

从微观和纳米物理到连续介质力学，动力学方程在数学物理的许多领域中发挥着核心作用。在物理和社会科学应用的数学描述中，它们是不可或缺的工具，从半导体、聚合物和等离子体到交通联网和蜂拥而至。这一提议的最终目标是开发新的分析和数值方法，其基础是具有多尺度和广泛应用的复杂现象的动力学描述。其目的是为了更好地理解当前研究的前沿问题，并通过理论、模型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-得克萨斯大学和法国图卢兹大学的顶尖研究人员领导，他们将结合他们在构建、分析和实施一系列选定问题的动力学描述方面的专业知识，从量子和微观尺度到宏观尺度。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。作为动力学方法论最近的一个新例子，我们将使用动力学描述来研究复杂供应链的双曲线流动。这一研究方案的理论和建模方面，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”。这一提议背后的理由是适时地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取得了许多重大进展，有可能对物理和社会科学中新兴的新领域产生相当大的影响。拟议中的努力将显著加强美国研究人员在从事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进行高级培训，并为与相关学科的研究人员进行跨学科互动提供平台，为科学队伍的发展做出贡献。将通过协同协作来保持内部和外部的互动，这些协作将在马里兰州(第一年)、法国和布朗(第二年)和威斯康星州(第三年)举行的三个年度讲习班期间取得成果。国际会议将作为马里兰大学科学计算和数学建模中心(CSCAMM)举办的一系列跨学科讲习班的一部分。项目研究人员将与美国能源部CSCAMM多尺度等离子体动力学中心、爱荷华州立大学能源部艾姆斯实验室和得克萨斯大学奥斯汀大学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.