Multiscale Computation in Kinetic Theory

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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.