Mean-Field Limits of Quantum Many-Body Dynamics and Free Boundaries in Kinetic Theory

量子多体动力学的平均场极限和运动理论中的自由边界

• 批准号: 1464869
• 负责人: Xuwen Chen
• 金额: $ 16.23万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1464869&HistoricalAwards=false
• 关键词: Mean; Field; Limits; Quantum; Many

This research program concerns the study of the rigorous justification of mean-field limits of quantum many-body dynamics and free boundaries problems in kinetic theory. Many-body systems arise naturally as fundamental models for physical systems. Since these many-body systems could contain 10^23 particles or more, the simulation of such systems is only possible via some approximation, such as the so-called mean-field limits. The mathematical justification of these mean-field limits, from the many-body systems they are supposed to describe, is therefore an issue of fundamental scientific importance. Two projects arise from the study of Bose-Einstein condensation (BEC). BEC is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. A large fraction of the bosons occupy the same quantum state, at which point quantum effects become apparent on a macroscopic scale. Since the Nobel-Prize-winning first observation of BEC in 1995, the investigation of this new state of matter has become one of the most active areas of contemporary research. Another project involves the determination of the motion of an object that is influenced by a sea of particles around it.The particular scope of this research project is to investigate several problems concerning the fine properties of solutions to the time-dependent many-body Schrödinger equation when the particle number tends to infinity and free boundary problems in kinetic theory. This research project encompasses three broad directions. The first direction concerns the space-time regularity of the solution to the BBGKY hierarchy under Gross-Pitaevskii scaling in three dimensions in the important case in which the scattering length of the microscopic interaction potential emerges. The second direction focuses on the derivation of focusing nonlinear Schrödinger equations from quantum many-body systems with focusing interactions. The third direction turns to the study of kinetic theory with boundaries and focuses on the effects on the asymptotic behaviors near equilibrium in caused by free boundaries. The PI and collaborators will use techniques from harmonic analysis, probability, and spectral theory to analyze these problems.

本研究计划关注量子多体动力学平均场极限的严格论证和动力学理论中的自由边界问题的研究。多体系统作为物理系统的基本模型自然出现。由于这些多体系统可能包含10^23个或更多的粒子，因此只能通过一些近似来模拟这样的系统，例如所谓的平均场极限。因此，这些平均场极限的数学证明，从它们应该描述的多体系统来看，是一个具有基本科学重要性的问题。两个项目源于玻色-爱因斯坦凝聚（BEC）的研究。玻色-爱因斯坦凝聚（BEC）是玻色子的稀气体冷却到非常接近绝对零度的物质状态。大部分玻色子占据相同的量子态，此时量子效应在宏观尺度上变得明显。自1995年诺贝尔奖获得者首次观测到BEC以来，对这种新物质状态的研究已成为当代研究中最活跃的领域之一。另一个研究项目是确定受周围粒子海影响的物体的运动。本研究项目的具体范围是研究与时间相关的多体薛定谔方程在粒子数趋于无穷大时的解的精细性质以及动力学理论中的自由边界问题。这个研究项目包括三个大方向。第一个方向涉及的空间-时间的规则性的解决方案的BBGKY层次下的Gross-Pitaevskii标度在三维中的重要情况下，其中出现的散射长度的微观相互作用潜力。第二个方向集中在从具有聚焦相互作用的量子多体系统导出聚焦非线性薛定谔方程。第三个方向转向研究有边界的动力学理论，重点研究自由边界对平衡点附近渐近行为的影响。PI和合作者将使用谐波分析，概率和频谱理论来分析这些问题。