Dynamical Processes in Many-Body Systems: Analysis and Simulations

许多相互作用电子的量子物理是化学和凝聚态物理的基础。直接处理多电子问题是不可能的，因为它的剪切复杂性：处理N个相互作用的电子需要求解3N维的偏微分方程组。平衡和非平衡密度泛函理论(DFT)是将相互作用的N电子问题转化为非相互作用的N电子问题的严格而形式上精确的理论。不相互作用的电子在有效势中运动，它与总电子密度有普遍的泛函依赖关系。结果，该问题被简化为3维的问题，易于计算。在这项建议中，PI建议在数学家和物理学家的交叉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，用于平衡和依赖于时间的情况。将进一步研究该理论及其解的数学结构，并将利用从该分析中获得的见解来开发有效的数值模拟。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，将特别强调自旋-轨道相互作用的处理。PI还计划建立耗散含时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，先进材料的设计包括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，拓宽有趣和潜在有用的材料阵列，并加快测试和表征的进程。这就是本文提出的研究目标。该计划是将严格的数学分析、物理、化学和计算机模拟的见解结合起来，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。拟议的研究可能会在纳米科学等应用领域以及太阳能电池设备和能量转换和储存等其他感兴趣的领域产生重大技术影响。个人投资促进机构建议通过让本科生和研究生以及博士后助理在跨学科环境中参与，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任职人数不足的群体中招聘妇女和学生。

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.