Mathematical Analysis of Interacting Quantum Gases

相互作用的量子气体的数学分析

• 批准号: 0652356
• 负责人: Robert Seiringer
• 金额: $ 12万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652356&HistoricalAwards=false
• 关键词: Mathematical; Analysis; Interacting; Quantum; Gases

The main focus of this research project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g., Bose-Einstein condensation and superfluidity, which makes them interesting both from a physical and a mathematical point of view.Intellectual Merit. From the point of view of mathematical physics, there has been substantial progress in the last few years in understanding some interesting phenomena occurring in quantum gases, and the goal of this project is to further investigate some of the relevant issues. Due to the complex nature of the problems, new mathematical ideas and methods will have to be developed for this purpose. Progress along these lines can be expected to yield valuable insight into the complex behavior of many-body quantum systems at low temperature. Among the questions that are addressed in this project are bounds on the free energy of quantum gases at low density and low temperature, as well as qualitative and quantitative statements about the corresponding thermalequilibrium states. The systems to be considered include both homogeneous and trapped systems, either continuous or on a lattice. The questions of interest concern, e.g., Bose/Fermi mixtures, low dimensional systems, rapidly rotating gases, as well as superfluidity for lattice systems. Moreover, the jellium model for gases of charged particles will be investigated, with the goal of further increasing the understanding of charged systems with Coulomb interaction.Broader Impact. The goal of this project is the development of new mathematicaltools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and can thus increase the understanding of physical systems. These methods will be useful, and will be used in physics graduate courses of the P.I. and others.

本课题的主要研究重点是多体量子系统的数学分析，特别是低温下量子气体的相互作用。最近在研究超冷原子气体方面的实验进展重新引起了人们对这些系统的兴趣。它们展示了丰富多样的量子现象，包括玻色-爱因斯坦凝聚和超流，这使得它们从物理和数学的角度都很有趣。从数学物理的角度来看，过去几年在理解量子气体中发生的一些有趣现象方面取得了实质性进展，该项目的目标是进一步研究一些相关问题。由于问题的复杂性，必须为此目的开发新的数学思想和方法。这些方面的进展有望为低温下多体量子系统的复杂行为提供有价值的见解。在这个项目中涉及的问题包括低密度和低温下量子气体的自由能的界限，以及关于相应的热平衡状态的定性和定量陈述。所考虑的系统既包括均匀系统，也包括连续的或格子上的捕获系统。感兴趣的问题包括玻色/费米混合物、低维体系、快速旋转的气体以及晶格体系的超流性。此外，还将研究带电粒子气体的凝胶模型，目的是进一步加深对具有库仑相互作用的带电系统的理解。这个项目的目标是开发新的数学工具来处理多体量子系统中的复杂问题。新的数学方法产生了不同的观点，因此可以增加对物理系统的理解。这些方法将是有用的，并将在P.I.和其他机构的物理研究生课程中使用。