Dynamical Processes in Many-Body Systems: Analysis and Simulations

许多相互作用的电子的量子物理学是化学和凝聚态物理的基础。由于剪切复杂性，直接处理多电子问题是不可能的：处理N个相互作用的电子需要在3N维上求解偏微分方程。平衡和非平衡密度泛函理论（DFT）是将相互作用n电子问题映射为非相互作用n电子问题的严格和形式精确的理论。非相互作用的电子以有效势运动，有效势与总电子密度具有普遍的函数依赖性。结果，该问题被简化为三维问题，便于计算。在本提案中，pi建议在数学家和物理学家的跨学科环境中研究多体量子力学中的一些动力学问题。特别是，pi建议进一步发展密度泛函理论的数学基础，用于平衡和时间相关的情况。该理论的数学结构及其解决方案将进一步研究，并从这一分析的见解将用于开发有效的数值模拟。将特别强调在完整的相对论公式和包含相对论修正的非相对论公式中对自旋轨道相互作用的处理。pi还计划建立耗散时变密度泛函理论的基础，并将该理论应用于材料中的电荷和自旋输运问题。目前的技术进步在很大程度上是基于新材料的设计和发现。如今，先进材料的设计涉及实验室工作和计算机模拟。提高计算机模拟的准确性和效率将降低成本，扩大有趣和潜在有用材料的阵列，并加快测试和表征的过程。这是本研究的目标。该计划将结合严格的数学分析，物理，化学和计算机模拟的见解，以推动纳米结构材料，拓扑绝缘体和分子电子器件等先进材料的理论模拟的边界。拟议的研究可能对纳米科学和其他感兴趣的领域（如太阳能电池装置和能量转换和存储）的应用产生重大的技术影响。pi建议在一个跨学科的环境中，通过让本科生、研究生和博士后参与进来，将研究和教育结合起来。将特别注意通过利用参与机构的各种方案，从其他代表性不足的群体中招收妇女和学生。

The quantum physics of many interacting electrons lies at the foundation of chemistry and condensed matter physics. A direct treatment of the many-electron problem is impossible due to its shear complexity: dealing with N interacting electrons requires solving partial differential equations in 3N dimensions. Equilibrium and non-equilibrium Density Functional Theories (DFT) are rigorous and formally exact theories which map the interacting N-electron problem into a non-interacting N-electron problem. The non-interacting electrons move in an effective potential that has a universal functional dependence on the total electron density. As a result, the problem is reduced to a problem in dimension 3, amenable for computation. In this proposal the PIs propose to study a number of dynamical problems in many-body quantum mechanics within an interdisciplinary environment of mathematicians and physicists. In particular, the PIs propose to develop further the mathematical foundations of density-functional theory, for equilibrium as well as the time-dependent case. The mathematical structure of the theory and its solutions will be further investigated and the insight from this analysis will be used to develop efficient numerical simulations. Particular emphasis will be given to the treatment of the spin-orbit interaction, within the full relativistic formulations and in non-relativistic formulations that include relativistic corrections. The PIs also plan to establish the foundations of the Dissipative Time-Dependent Density Functional Theory, and to apply the theory to the problem of charge and spin transport in materials.The present technological progress is in great part based on design and discovery of new materials. Nowadays, the design of advanced materials involves laboratory work and computer simulations. Enhancing the accuracy and efficiency of computer simulations will reduce the costs, broaden the array of interesting and potentially useful materials, and speed up the process of testing and characterization. This is the target of the proposed research. The plan is to combine rigorous mathematical analysis, the insights from physics, chemistry and computer simulations in order to push the boundaries of theoretical simulations of advanced materials such as nano-structured materials, topological insulators and molecular electronic devices. The proposed research could have significant technological impact in applications such as nano-science and other areas of interest such as solar cell devices and energy conversion and storage. The PIs propose to integrate research and education by involving undergraduate and graduate students, and post-doctoral associates, in an interdisciplinary environment. Special attention will be paid to the recruitment of women and students from other underrepresented groups through the utilization of a diverse number of programs at the participating institutions.