Interacting Particle Systems and Nonlinear Partial Differential Equations

分析相互作用粒子的大系统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（理解玻色子星）到社会研究（建模社会网络）。由于粒子的数量通常非常大，人们希望通过一些宏观的平均特征来理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，研究粒子数趋近于无穷时的渐近行为是有帮助的，并假设该极限近似于具有有限粒子数的大系统所观察到的性质。描述大型粒子系统的宏观行为的一个重要现象的例子是玻色-爱因斯坦凝聚（BEC），这是稀释玻色气体在极低温度下作为单个粒子运动时的物质状态。尽管玻色和爱因斯坦在量子力学的早期就预测到了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）。人们建立了数学模型来理解这种现象。这些模型将相互作用粒子的大量子系统和非线性偏微分方程（PDE）联系起来，这些方程是在粒子数量趋于无穷的极限下由这些系统导出的。然而，两端仍然存在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，首席研究员将致力于解决这些问题。研究小组将继续探索不同的途径，透过设计和教授新课程（例如为暑期学校的研究生设计和教授多门课程）、培训和指导研究生和博士后，以及组织和参加研讨会和研究会议，传播从建议项目中获得的知识。随着从量子多体系统（如非线性薛定谔方程）和经典多粒子系统（如玻尔兹曼方程）推导有效方程的基础工作的开展，开辟了数学物理和非线性偏微分方程界之间交流的新渠道，促进了这两个领域的发展。特别是近年来，从相互作用玻色子的量子系统中严格推导非线性薛定谔方程取得了显著的进展。在这一进展的激励下，大约十年前，PI和她的同事Chen开始研究量子多粒子系统和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，通过起源于1粒子非线性PDE（即NLS）背景的思想和技术，开发了研究量子多粒子系统的程序。在当前的项目中，PI和合作者将显著扩展上述计划的范围，包括：非线性偏微分方程的定性方面的推导，例如哈密顿量或可积，以及经典粒子系统的分析，从而导致新的动力学方程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.