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

这项研究计划涉及到量子多体动力学平均场极限的严格证明和运动论中的自由边界问题。多体系统自然而然地成为物理系统的基本模型。由于这些多体系统可能包含10^23个或更多的粒子，因此只能通过某种近似来模拟此类系统，例如所谓的平均场极限。因此，从它们应该描述的多体系统出发，对这些平均场极限进行数学证明，这是一个具有基础性科学意义的问题。对玻色-爱因斯坦凝聚(BEC)的研究产生了两个项目。BEC是玻色子稀薄气体的一种物质状态，冷却到非常接近绝对零度的温度。很大一部分玻色子占据相同的量子态，在这一点上，量子效应在宏观尺度上变得明显。自从1995年诺贝尔奖获得者首次观测到BEC以来，对这种新的物质状态的研究已经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粒子海影响的对象的运动。该研究项目的具体范围是研究与含时多体薛定谔方程的解的精细性质有关的几个问题，当粒子数趋于无穷大时，以及动力学理论中的自由边界问题。这项研究项目包括三个大方向。第一个方向是在微观相互作用势的散射长度出现的重要情况下，三维Gross-Pitaevskii标度下BBGKY方程解的时空正则性。第二个方向是从具有聚焦相互作用的量子多体系统中推导出聚焦非线性薛定谔方程。第三个方向转向有边界的动力学理论的研究，重点是自由边界对平衡点附近渐近行为的影响。PI和合作者将使用调和分析、概率和频谱理论的技术来分析这些问题。

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.