Kinetic Transport of Interactive Complex Particle Dynamics in Mean Fields.

University Of Texas At Austin, Austin TX

Investigators

Abstract

The overall objective of the principal investigator's (PI) research is to develop accurate mathematical models and computer simulations arising from physical phenomena of fundamental scientific interest. The mathematical problems considered in this project describe non-equilibrium systems endowed with memory effects. Such systems are characterized by internal and external forces that generate breaking of symmetry and exhibit stable states that cannot be captured by simple hydrodynamics within classical fluid and gas dynamics modeling. The elements of these systems arise in many phenomena impacting daily human life: e.g., biosystems and molecular medicine at miniature scale, plasma evolution in fusion models for clean energy, and reacting solid state nano structures for solar generation of hydrogen resources, to name just a few. The broad range of problems requires new computational approaches that are being designed and analyzed within this project to ensure consistency, stability, error estimates control and rates of convergence to equilibrium. The scientific computing component is being developed using the techniques that need to be integrated into novel AI and ML strategies along with the tools from non-linear analysis. The work is interdisciplinary in nature and is being carried out in collaboration with physicists, engineers, and social scientists. The PI’s students and postdoc trainees will be involved in the research. These research goals comprise a broad program in the development of analytical and numerical tools associated with statistical transport equations and systems at the core of applied mathematics in probability, statistics applied to chemistry, physics as well as biological and social dynamics. They concern the modeling of complex interactions systems yielding kinetic frameworks associated to Markovian and non-Markovian processes of birth-death dynamics such as Chapman-Kolmogorov flows of weak turbulence problems arising as dissipative mechanisms in Vlasov-Poisson or Maxwell systems. Such statistical approaches lead to nonlinear integro-differential systems of equations of collisional classical, or quantum Boltzmann of Dirac-Fermi or Bose Einstein type, or aggregation, coalescence, breakage particle systems. The PI will focus on the interplay of this models from analytical and numerical mathematics viewpoint and the scientific computing implementations. 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.