Many-Body Dynamics and Nonlinear Evolution Equations

了解相互作用的大粒子系统的行为对于预测和理解从物理学中的恒星结构到社会网络的动力学等各种背景下出现的现象至关重要。由于这类模型中的粒子数量非常多，人们希望通过宏观的平均特性来理解这类系统的定性和定量性质。这类似于通过温度和压力来描述气体体积中巨大数量的气体原子。为了识别多粒子系统的宏观行为，研究当粒子数趋于无穷大时的渐近行为是很有帮助的，希望这个极限将接近于在粒子数很大但有限的系统中观察到的性质。这种宏观行为描述的一个重要现象的例子是大粒子系统的玻色-爱因斯坦凝聚(BEC)，这是一种稀薄玻色气体在极低温度下的物质状态。为了理解这一现象而开发的数学模型将相互作用的粒子组成的大型量子系统与由此类系统导出的非线性偏微分方程(PDE)联系在一起，这些系统在粒子数量无限大的情况下得到了限制。非线性偏微分方程组理论正在成为理解这些极限的重要数学机制。这一研究项目致力于在相互作用粒子的复杂系统和非线性PDE之间建立进一步的联系，包括已被提出作为许多基本波动现象的模型的波动和色散方程，从玻色-爱因斯坦凝聚到海洋中畸形波的形成，以及描述稀薄气体动力学的动力学方程，它们是应用分析、概率和统计物理的核心。研究活动包含一种跨学科的方法，有可能将事实证明在非线性偏微分方程水平上有用的想法和技术应用于许多身体系统的分析，同时结合数学分析、概率和统计物理的各种工具。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)导出有效方程的基础工作的开展，数学物理和非线性偏微分方程组之间的一条新的交流渠道已经打开，为这两个领域的发展做出了贡献。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之间的新出现的联系。PI计划进一步扩展这项工作，以探索从许多物体粒子系统推导齐次玻尔兹曼方程的某些方面。PI将研究Boltzmann方程的分析特性，Boltzmann方程是一个统计相互作用粒子系统流的概率模型，其中含有一个非线性项，涉及一个解释碰撞或相互作用速率的碰撞核。由于Boltzmann方程的解是概率分布的，研究其多项式矩和指数矩的行为是一个基本的分析工具。特别地，由于玻尔兹曼方程的平衡态是高斯分布，人们预计解将受控于指数衰减界。因此，还将研究解的指数矩的界。

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.