FRG: Collaborative Research: Dynamical Processes in Many-Body Systems: Analysis and Simulations

FRG：协作研究：多体系统中的动态过程：分析和仿真

• 批准号: 1065894
• 负责人: Weinan E
• 金额: $ 62.26万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1065894&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Dynamical; Processes; FRG; Collaborative; Research; Dynamical; Processes

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.

许多相互作用的电子的量子物理学在于化学和凝结物理学的基础。由于其剪切复杂性，无法直接处理多电子问题：处理n个相互作用的电子需要在3n维度中求解部分微分方程。平衡和非平衡密度函数理论（DFT）是严格且正式精确的理论，可将相互作用的N电子问题映射到非相互作用的N电子问题中。非相互作用电子以有效电位具有对总电子密度具有通用功能依赖性的有效电势。结果，该问题被简化为“维度3”中的问题，可用于计算。在此提案中，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的许多动态问题。特别是，PI提议进一步发展密度功能理论的数学基础，以达到平衡以及时间依赖性情况。该理论及其解决方案的数学结构将得到进一步研究，该分析的洞察力将用于开发有效的数值模拟。在完整的相对论表述中以及包括相对论校正的非相对论表述中，将特别强调自旋轨道相互作用。 PI还计划建立耗散时间依赖性密度功能理论的基础，并将理论应用于材料中的电荷和旋转运输问题。当前的技术进步在很大程度上是基于设计和发现新材料的。如今，高级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，扩大一系列有趣且可能有用的材料，并加快测试和表征的过程。这是拟议研究的目标。该计划是结合严格的数学分析，物理，化学和计算机模拟的见解，以突破高级材料的理论模拟的边界，例如纳米结构材料，拓扑绝缘子和分子电子设备。拟议的研究可能会对纳米科学和其他感兴趣领域（例如太阳能电池设备以及能量转换和存储）等应用产生重大的技术影响。 PIS建议通过在跨学科环境中涉及本科生和研究生以及博士后同事来整合研究和教育。通过利用参与机构的各种计划，将特别注意从其他代表性不足的群体中招募妇女和学生。