Multiscale Computation in Kinetic Theory

物理系统在不同的尺度上被建模，并使用不同的方程进行不同的近似。量子水平上的薛定谔方程模拟了分子和原子尺度。 经典力学中的牛顿定律是宏观尺度的模型。 玻尔兹曼方程适用于统计水平，其中研究了许多粒子的系统，Navier-Stokes方程和其他模型分布系统，如连续体系中的流体。应用数学和物理学的一个中心问题是理解不同模型之间的关系，多年来已经开发了许多工具（包括分析和数值）来完成这一任务。然而，他们中的大多数理想化的系统正在研究中，并不能解决实际问题，已经出现在化学，物理学和工程复杂系统的研究。该项目的重点是两个长期存在的挑战，这些连接：量子信息的表征在经典制度时，化学反应的存在，以及统计和流体描述之间的耦合。该项目旨在开发多尺度系统建模的改进方法。 尽管它们在物理学和工程学中具有根本的重要性，但动力学理论中多尺度问题的有效数学分析和计算技术仍然相当难以捉摸。许多物理系统中固有的多尺度已经提出了臭名昭著的计算挑战。 该项目涉及动力学理论中多尺度数值方法的发展，包括Boltzmann型方程的流体动力学极限和Schrodinger方程的半经典极限的数值捕获。这两个问题长期以来一直被认为是动力学理论中的基本问题。更具体地说，该项目的重点是捕捉来自量子力学的经典区域中的非绝热转变，以及将流体描述与统计力学描述联系起来的边界层效应。这两个问题都出现在过渡区，多物理现象可以捕捉到目前可用的数学处理，计算是远远不够的效率。该项目旨在为这些问题开发和分析有效的计算工具，重点是边界层和界面的处理以及渐近保持方案的设计。 除了导致这些类型的物理系统的更好的理解，预计正在开发的新工具可以激发在其他领域出现的类似问题的治疗，例如，随机介质的双曲型问题。

Physical systems are modeled at different scales and to different approximations with different equations. The Schrodinger equation at the quantum level models the molecular and atomic scale. Newton's laws in classical mechanics model the macroscopic scale. The Boltzmann equation applies at the statistical level, where systems of many particles are studied, and the Navier-Stokes equation and others model distributed systems such as fluids in the continuum regime. A central question in applied mathematics and physics is to understand the relationships between the different models, and many tools (both analytical and numerical) have been developed for this task through the years. However, most of them idealize the systems under study and cannot tackle practical problems that have emerged in the study of complicated systems in chemistry, physics, and engineering. This project focuses on two longstanding challenges concerning these connections: the characterization of quantum information in the classical regime when chemical reactions are present, and the coupling between the statistical and the fluid description. The project aims to develop improved methods for the modeling of multiscale systems. Despite their fundamental importance in physics and engineering, effective mathematical analysis and computational techniques for multiscale problems in kinetic theory have remained rather elusive. The multiple scales inherent in many physical systems have posed notorious computational challenges. This project concerns development of multiscale numerical methods in kinetic theory, including numerical capture of the hydrodynamic limit of Boltzmann-type equations and the semi-classical limit of the Schrodinger equation. Both have long been regarded as fundamental problems in kinetic theory. More specifically, the project focuses on capturing the non-adiabatic transition in the classical regime derived from quantum mechanics, and boundary layer effects that connect the fluid description with the statistical mechanical description. Both problems emerge in transition regimes, the multi-physics phenomena can be captured by none of currently available mathematical treatments, and the computation is far from being efficient. The project aims to develop and analyze efficient computational tools for these problems, focusing on treatment of boundary layers and interfaces and design of asymptotic-preserving schemes. Besides leading to improved understanding of physical systems of these types, it is expected that the new tools under development could inspire treatments of similar problems emerging in other areas, for example, hyperbolic type problems with random media.