Dynamical Processes in Many-Body Systems: Analysis and Simulations

许多相互作用电子的量子物理学是化学和凝聚态物理学的基础。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在3 N维中求解偏微分方程。平衡态和非平衡态密度泛函理论（DFT）是严格的和形式上精确的理论，它将相互作用的N-电子问题映射为非相互作用的N-电子问题。非相互作用的电子在有效势中移动，该有效势对总电子密度具有普适的函数依赖性。结果，问题被简化为三维问题，适合计算。在这个提议中，PI提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，平衡以及时间依赖的情况下。理论的数学结构及其解决方案将被进一步研究，从这种分析的见解将被用来开发有效的数值模拟。特别强调的是，将给予治疗的自旋轨道相互作用，在完整的相对论制剂和非相对论制剂，包括相对论修正。他们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范围，并加快测试和表征过程。这是拟议研究的目标。该计划将联合收割机结合严格的数学分析，物理，化学和计算机模拟的见解，以推动先进材料，如纳米结构材料，拓扑绝缘体和分子电子器件的理论模拟的界限。拟议的研究可能会对纳米科学等应用以及太阳能电池设备、能量转换和储存等其他感兴趣的领域产生重大的技术影响。PI建议通过在跨学科环境中让本科生和研究生以及博士后同事参与，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代表性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.