Simulation of the Wave-Matter Interactions at Extreme Scales

本项目致力于为在几何光学、动力学模型和纳米光学等极端尺度下研究波-物质相互作用提供可靠的模型和最先进的数值方法，并在天体物理学、地球科学、等离子体物理学、聚变能科学、生物学、半导体技术、材料科学、纳米技术和纳米科学等领域得到实际应用。例如，地球物理勘探过程，核聚变能源科学中的波等离子体相互作用，以及用于太阳能转换的纳米电机的制造。所提出的模型和数值方法是基于几何光学、物理光学、动力学理论、纳米光学、计算化学和科学计算的知识。将进行跨学科合作，以扩大所建议方法的实际应用。该项目还涉及计算数学的研究和教育的整合。我们鼓励研究生和本科生以及来自代表性不足群体的成员参与该项目，以提高他们的知识和研究水平。该计划旨在进一步培训和教育学生，并鼓励他们在未来从事科学、技术、工程和数学（STEM）方面的职业。在本项目中，将提出可靠的模型和有效的数值方法来模拟极端尺度下的波-物质相互作用：(1)如果物质的大小远远大于波长，则相互作用相当于非均匀介质中的高频波传播。PI将开发结合波的积分表示和渐近高频理论（特别是几何光学）的渐近方法。（II）如果物质的大小减小到中观或微观尺度，则普遍采用动力学模型来模拟相互作用。PI将开发利用6维概率分布函数的Hopf-Cole变换的渐近方法。（三）如果物质的大小不断减小到纳米或原子尺度，则物质的运动必须用量子力学来确定，而波的传播则必须用量子力学来确定。PI将使用半经典理论作为构建块，以开发数字可跟踪的半经典模型和有效的多尺度方法。核心思想包括：对于(1)，用格林函数的积分表示将作为波的传播机制，几何光学近似将提供格林函数的信息，并设计快速多级算法以最优复杂性高效地计算振荡积分。对于（II），将概率分布函数的Hopf-Cole变换的相函数近似为幂级数展开式，展开式项由在三维空间空间中表述的方程的解确定，可以设计高阶格式来高效地求解此类方程。对于（III），波的传播将通过经典的麦克斯韦方程来确定，Ehrenfest分子动力学和时变电流密度泛函数理论将被应用于解决处理多体薛定谔方程的困难，并且模型将通过精心设计的多尺度求解器来求解。

This project is devoted to providing reliable models and the state-of-the-art numerical methods for studying the wave-matter interactions at extreme scales as in geometrical optics, kinetic models, and nano optics, with practical applications arising from astrophysics, geosciences, plasma physics, fusion energy sciences, biology, semiconductor technology, material sciences, nano-technology and nano-sciences. For example, the geophysical exploration processes, the wave-plasma interactions in fusion energy sciences, and the fabrication of nano-motors for solar energy conversion. The proposed models and numerical methods are based on knowledge of geometrical optics, physical optics, kinetic theories, nano optics, computational chemistry and scientific computing. Interdisciplinary collaborations will be pursued to extend the practical applications of the proposed methods. The project also involves the integration of research and education in computational mathematics. Graduate and undergraduate students, and members from underrepresented groups will be encouraged to participate in the project to enhance their knowledge and research. The proposed project will further the training and education of students and encourage them to pursue future career in science, technology, engineering and mathematics (STEM).In the project, reliable models and efficient numerical methods will be proposed to simulate the wave-matter interactions at extreme scales: (I) If the size of the matter is much greater than the wavelength, the interactions are equivalent to high frequency wave propagation in inhomogeneous media. The PI will develop asymptotic methods that combine integral representations of the waves and asymptotic high frequency theories (notably geometrical optics). (II) If the size of the matter decreases to meso or micro scale, kinetic models are popularly applied for modeling the interactions. The PI will develop asymptotic methods that utilize the Hopf-Cole transformation of the 6-D probabilistic distribution function. (III) If the size of the matter keeps decreasing to nano or atomic scale, the motion of the matter must be determined quantum mechanically while the wave propagation is determined. The PI will use semi-classical theories as the building block to develop numerically trackable semi-classical models and efficient multi-scale methods. The core ideas consist of the following: For (I), integral representations with Green's functions will serve as the mechanism for wave propagation, geometrical optics approximations will provide the information of Green's functions, and fast multi-level algorithms will be designed to evaluate the oscillatory integrals efficiently with optimal complexities. For (II), the phase function of the Hopf-Cole transformation of the probabilistic distribution function will be approximated as a power series expansion with the expansion terms determined through solutions of equations formulated in 3-D spatial space, and high-order schemes can be designed to solve such equations efficiently. For (III), the wave propagation will be determined classically through Maxwell's equations, Ehrenfest molecular dynamics and time-dependent current density functional theory will be applied to resolve the difficulty of dealing with many-body Schrodinger equations for the matter, and the models will be solvable by well-designed multi-scale solvers.