Interacting Particle Systems and Nonlinear Partial Differential Equations

对相互作用的大粒子系统的分析是预测和理解在不同背景下出现的各种现象的关键，从物理学(理解玻色子恒星)到社会研究(在对社会网络建模时)。由于粒子的数量通常很大，人们希望通过一些宏观的、平均的特征来了解这种粒子系统的定性和定量性质。为了识别多粒子系统的宏观行为，在假设极限将近似于具有大量有限粒子的系统中观察到的性质的前提下，研究粒子数趋于无穷大时的渐近行为是很有帮助的。描述大粒子系统宏观行为的一个重要现象是玻色-爱因斯坦凝聚(BEC)，它是稀薄玻色气体在极低温度下作为单个粒子运动时的一种状态。尽管玻色和爱因斯坦在量子力学的早期就预言了BEC，但第一次实验实现是在1995年(随后在2001年获得了诺贝尔物理学奖)。为了理解这种现象，已经建立了数学模型。这些模型将相互作用的粒子组成的大型量子系统与从这些系统派生出来的非线性偏微分方程(PDE)联系在一起，这些系统的粒子数量有限。然而，两端仍有许多具有挑战性的问题，这些问题可以从跨学科的角度受益，首席调查员将致力于解决这些问题。主办方将继续探索通过设计和教授新课程(例如，主办方为暑期学校的研究生设计和教授多个课程)、培训和指导研究生和博士后，以及通过组织和参加研讨会和研究会议，传播从拟议项目中获得的知识的不同方式。通过从量子多体系统(如非线性薛定谔方程)和经典多粒子系统(如Boltzmann方程)导出有效方程的基础工作，开辟了数学物理和非线性PDE社区之间的新的交流渠道，为这两个领域的进步做出了贡献。特别是，最近在从相互作用玻色子的量子系统严格推导非线性薛定谔(NLS)方程方面取得了显著的进展。在这一进展的推动下，大约十年前，Pi和她的同事Chen开始研究量子多粒子系统与NLS方程之间的联系，因此，他们与他们的合作者(包括11名博士生和3名博士后)一起开发了一个项目，通过起源于单粒子非线性PDE(即NLS)的思想和技术来研究量子多粒子系统。在目前的项目中，PI和合作者将显著扩展上述计划的范围，包括：非线性偏微分方程组的定性方面的推导，例如哈密顿或可积，以及导致新的动力学方程的经典粒子系统的分析。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Analysis of large systems of interacting particles is key for predicting and understanding various phenomena arising in different contexts, from physics (in understanding e.g. boson stars) to social studies (when modeling social networks). Since the number of particles is usually very large one would like to understand qualitative and quantitative properties of such systems of particles through some macroscopic, averaged characteristics. In order to identify macroscopic behavior of multi-particle systems, it is helpful to study the asymptotic behavior when the number of particles approaches infinity, with the assumption that the limit will approximate properties observed in the systems with a large finite number of particles. An example of an important phenomenon that describes such macroscopic behavior of a large system of particles is the Bose-Einstein condensation (BEC), which is a state of the matter of a dilute Bose gas at very low temperatures when the gas moves as a single particle. Although the BEC was predicted in early days of quantum mechanics by Bose and Einstein, the first experimental realization came in 1995 (subsequently recognized by a Nobel Prize in physics in 2001). Mathematical models have been developed to understand such phenomena. Those models connect large quantum systems of interacting particles and nonlinear partial differential equations (PDE) that are derived from such systems in the limit of the number of particles going to infinity. However there are still many challenging problems on both ends, that could benefit from an interdisciplinary perspective, and the Principal Investigator will work on these. The PI will continue to explore diverse ways to disseminate the knowledge obtained from the proposed projects via designing and teaching new courses (e.g. the PI designed and taught multiple courses for graduate students at summer schools), training and mentoring graduate students and postdocs, and via organizing as well as attending seminars and research meetings.problems. With fundamental works on derivation of effective equations from quantum many body systems (e.g. nonlinear Schrodinger equation) and effective equations from classical many particle systems (e.g. Boltzmann equation) a new channel of communication between mathematical physics and nonlinear PDE communities has opened, contributing to advances in both areas. In particular, recently remarkable progress has been achieved in the rigorous derivation of nonlinear Schroedinger (NLS) equations from quantum systems of interacting bosons. Motivated by that progress, about a decade ago, the PI and her colleague Chen started studying connections between quantum many particle systems and NLS equation, and consequently together with their collaborators (including 11 PhD students and 3 postdocs) they developed the program of studying quantum many particle systems via ideas and techniques that originated in the context of 1 particle nonlinear PDE, namely the NLS. In the current project, the PI and collaborators will significantly expand the span of the above program to include: derivation of qualitative aspects of nonlinear PDE, such as being Hamiltonian or integrable, and Analysis of classical systems of particles that lead to new kinetic equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.