Many-Body Dynamics and Nonlinear Evolution Equations

了解大型相互作用粒子系统的行为对于预测和理解各种背景下出现的现象至关重要，从物理学中的恒星结构到社会网络动力学。由于这种模型中的粒子数量非常大，人们希望通过宏观平均特征来理解这种系统的定性和定量性质。这类似于通过温度和压力对气体体积中大量气体原子的描述。为了识别多粒子系统的宏观行为，研究粒子数量趋于无穷时的渐近行为是有帮助的，希望该极限能近似于粒子数量大但有限的系统中所观察到的性质。这种宏观行为描述的一个重要现象的例子是一个大粒子系统的玻色-爱因斯坦凝聚（BEC），这是一种在极低温度下稀释玻色气体的物质状态。为了理解这一现象而开发的数学模型将相互作用的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大的极限下由这些系统导出的。非线性偏微分方程理论正在成为理解这种极限的重要数学机制。该研究项目的重点是建立相互作用粒子的复杂系统和非线性偏微分方程之间的进一步联系，包括波和色散方程，这些方程已被提出作为许多基本波现象的模型，从玻色-爱因斯坦凝聚到海洋中异常波的形成，以及描述稀气体动力学的动力学方程，这些方程是应用分析，概率和统计物理的核心。该研究活动包含一种跨学科的方法，有可能带来在非线性偏微分方程水平上对许多身体系统分析有用的想法和技术，同时结合了数学分析、概率和统计物理等多种工具。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Vlasov方程）推导有效方程的基础工作，开辟了数学物理和非线性PDE社区之间交流的新渠道，促进了这两个领域的进步。首席研究员将继续合作，了解许多身体系统的动力学和非线性PDE之间的新联系。PI进一步计划扩展这条工作线，探索从许多体粒子系统推导齐次玻尔兹曼方程的某些方面。PI将研究玻尔兹曼方程的分析性质，玻尔兹曼方程是相互作用粒子系统统计流的概率模型，具有非线性项，涉及碰撞核，该项说明碰撞或相互作用的速率。鉴于玻尔兹曼方程的解是概率分布，研究其多项式矩和指数矩的行为是一种基本的分析工具。特别地，由于玻尔兹曼方程的平衡态是高斯分布，人们期望它的解会被指数衰减的边界所控制。因此，解的指数矩的界也将被研究。

Understanding behavior of large systems of interacting particles is essential for predicting and understanding phenomena arising in various contexts, from stellar structure in physics to dynamics of social networks. Since the number of particles in such models is very large, one would like to understand qualitative and quantitative properties of such systems through macroscopic, averaged characteristics. This is similar to the description of the huge number of gas atoms in a volume of gas through temperature and pressure. 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 hope that the limit will approximate properties observed in the systems with a large, but finite, number of particles. An example of an important phenomenon described by such macroscopic behavior is the Bose-Einstein condensation (BEC) of a large system of particles, which is a state of the matter of a dilute Bose gas at very low temperatures. Mathematical models developed to understand the phenomenon 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. The theory of nonlinear PDE is becoming an important mathematical mechanism in understanding such limits. This research project focuses on establishing further connections between complex systems of interacting particles and nonlinear PDE, including wave and dispersive equations that have been proposed as models for many basic wave phenomena, from Bose-Einstein condensation to formation of freak waves in an ocean, and kinetic equations that describe dynamics of a dilute gas and are at the core of applied analysis, probability, and statistical physics. The research activity contains an interdisciplinary approach with potential to bring ideas and techniques that turned out to be useful at the level of a nonlinear PDE to the analysis of many body systems, while incorporating diverse tools from mathematical analysis, probability, and statistical physics. With fundamental work 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. Vlasov equation) a new channel of communication between mathematical physics and nonlinear PDE communities has opened, contributing to advances in both areas. The Principal Investigator will continue collaborative work on understanding emerging connections between dynamics of many body systems and nonlinear PDE. The PI further plans to expand this line of work to explore some aspects of a derivation of the homogeneous Boltzmann equation from many body particle systems. The PI will study analytical properties of the Boltzmann equation, a probabilistic model for a statistical flow of interacting particle systems with a nonlinear term involving a collision kernel that accounts for the rate of collisions or interactions. Given that the solutions of the Boltzmann equation are probability distributions, the study of the behavior of its polynomial and exponential moments is a fundamental analytical tool. In particular, since the equilibrium state for the Boltzmann equation is a Gaussian distribution, one expects that the solution would be controlled by bounds of exponential decay. Thus, bounds for exponential moments of the solution will be studied as well.