Non-local Kinetic Collisional Transport: Analysis and Numerical Methods

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The overall objective of this research is to develop accurate mathematical modeling and methods of numerical simulation for an array of diverse natural and engineering processes of fundamental scientific interest, such as evolution of plasmas in fusion devices, dynamics of very cold gases in the intermediate transition to form Bose-Einstein condensates,  hot-electron transport in semiconductor devices, design of nanostructures for the solar generation of hydrogen, dynamics of reacting molecular mixtures associated to aerospace dynamics of re-entry problems. All these phenomena are unified by the underlying structure of their mathematical description that involves "particles" interactions of different kinds. A study of these mathematical structures is the principal subject of this project. Modelling and simulation will be based on data obtained by accurate crystallographic calculations, taking into account atomistic corrections, the presence of rough media, etc. Same of the techniques that we will develop are pertinent to exciting new applications in biological and social sciences. They include modeling of self-organized flows in "particle" swarms like birds or fish, emerging consensus in population dynamics, multi-agent information transfer and social information dynamics in internet, to name a few. Research goals of the project comprise a broad program in the development of analytical and numerical tools associated with statistical transport equations at the core of applied mathematics in probability, statistics applied to chemistry, physics and to an extent, to social dynamics as well. They concern the modeling of complex interactions systems yielding kinetic frameworks associated to Markovian processes of birth-death dynamics. Such statistical approaches lead to nonlinear integro-differential systems of equations of collisional classical or quantum Boltzmann or Smoluchowski type. Many of these models appear in the collisional theory of semi-classical transport for short- and long-range particle interactions models that describing self-consistent phenomena at nano and mesoscales.  New tools from nonlinear analysis as well as new computational strategies will be developed to address long-time behavior, stability and decay rates to stationary modes, as well as qualitative behavior of numerical solutions and optimal computational strategies.