Large Systems of Interacting Particles and Waves and Their Effective Equations

Large systems of interacting particles arise in different contexts, from physics (in understanding e.g. boson stars) to social studies (when modeling social networks). ). Since the number of particles is often very large, rather than analyzing each particle, 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 hope that the limit - which is typically described by a nonlinear partial differential equation (PDE) - 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). Since then mathematical models have been developed to understand such phenomena. Those models connect large quantum systems of interacting particles and nonlinear PDE that are derived from such systems. In this project, the principal investigator (PI) and her team will work on advancing such models and developing new ones by relying on properties of underlying physical systems and on mathematical understanding of nonlinear PDE. In particular, recently remarkable progress has been achieved in understanding how the dynamics of various nonlinear PDE such as, e.g. nonlinear Schrodinger equation or Boltzmann equation arise from large systems of interacting particles. However, there are still many open questions that are motivated by novel analytical studies of these effective nonlinear PDE or directly by physics experiments. The PI addresses some of these questions and continues her work on advancing the connections between large systems of interacting particles/waves on one side and effective equations that stem from these systems upon certain averaging procedures. In particular, the PI and her team will focus on exploring origins of certain geometric and algebraic properties of effective equations, pursue a mathematical study of mixtures of bosons and fermions (which is directly inspired by recent physics experiments), and study derivation and analytic properties of wave kinetic equations that play role in wave turbulence. 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.