Mean-Field Limits of Quantum Many-Body Dynamics and Free Boundaries in Kinetic Theory

University Of Rochester, Rochester NY

Investigators

Abstract

This research program concerns the study of the rigorous justification of mean-field limits of quantum many-body dynamics and free boundaries problems in kinetic theory. Many-body systems arise naturally as fundamental models for physical systems. Since these many-body systems could contain 10^23 particles or more, the simulation of such systems is only possible via some approximation, such as the so-called mean-field limits. The mathematical justification of these mean-field limits, from the many-body systems they are supposed to describe, is therefore an issue of fundamental scientific importance. Two projects arise from the study of Bose-Einstein condensation (BEC). BEC is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. A large fraction of the bosons occupy the same quantum state, at which point quantum effects become apparent on a macroscopic scale. Since the Nobel-Prize-winning first observation of BEC in 1995, the investigation of this new state of matter has become one of the most active areas of contemporary research. Another project involves the determination of the motion of an object that is influenced by a sea of particles around it. The particular scope of this research project is to investigate several problems concerning the fine properties of solutions to the time-dependent many-body Schrödinger equation when the particle number tends to infinity and free boundary problems in kinetic theory. This research project encompasses three broad directions. The first direction concerns the space-time regularity of the solution to the BBGKY hierarchy under Gross-Pitaevskii scaling in three dimensions in the important case in which the scattering length of the microscopic interaction potential emerges. The second direction focuses on the derivation of focusing nonlinear Schrödinger equations from quantum many-body systems with focusing interactions. The third direction turns to the study of kinetic theory with boundaries and focuses on the effects on the asymptotic behaviors near equilibrium in caused by free boundaries. The PI and collaborators will use techniques from harmonic analysis, probability, and spectral theory to analyze these problems.