Interacting Particle Systems and Nonlinear Partial Differential Equations

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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.