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.

该研究计划涉及对动力学理论中量子多体动力学和自由边界问题的平均场限制的严格理由的研究。多体系统自然是物理系统的基本模型。由于这些多体系统可能包含10^23个或更多粒子，因此只能通过一些近似（例如所谓的平均场限制）对这种系统进行仿真。因此，这些平均场限制的数学正当理由来自他们期望描述的多体系统，这是一个基本的科学重要性问题。两个项目来自Bose-Einstein凝结的研究（BEC）。 BEC是一种稀释气体的问题，冷却至非常接近绝对零的温度。大部分玻色子占据相同的量子状态，此时量子效应在宏观尺度上变得显而易见。自从1995年对诺贝尔奖赢得BEC的首次观察以来，对这种新物质状态的调查已成为当代研究中最活跃的领域之一。另一个项目涉及确定受周围颗粒海洋影响的物体运动的确定。该研究项目的特殊范围是研究有关粒子趋于粒子趋于无限性和动力学理论的无限边界问题时，有关解决方案的良好特性的几个问题。该研究项目包括三个广泛的方向。第一个方向涉及在三个维度缩放下缩放的BBGKY层次结构的解决方案的时空规则性，在重要情况下，微观相互作用势的散射长度出现了。第二个方向集中于从量子多体系统中聚焦非线性schrödinger方程的推导，并具有聚焦相互作用。第三个方向转向了具有边界的动力学理论的研究，并着重于自由边界引起的平衡附近的不对称行为的影响。 PI和合作者将使用谐波分析，概率和光谱理论中的技术来分析这些问题。