Many-Body Dynamics and Nonlinear Evolution Equations

University Of Texas At Austin, Austin TX

Investigators

Abstract

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.