The Kinetics of Interacting Particle Systems: Theory and Numerical Methods

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The overall objective of this research is to develop accurate modeling and simulation for a series of diverse phenomena of fundamental scientific interest, at the edge of various technological developments such as hot-electron transport in semiconductor devices, nano structures for the solar generation of hydrogen, reacting molecular mixtures, and evolution of plasmas in fusion models. Modeling and simulation will be based on data obtained by accurate crystallographic calculations, taking into account atomistic corrections, the presence of rough media, etc. Many of the techniques to be developed are pertinent also 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.These research goals 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, and physics. 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 Boltzmann or Smolukowski type. Many of these models appear in the collisional theory of semi-classical transport for short and long range interactions models that describe self-consistent phenomena at nano and mesoscales. More recently, collisional of birth-death processes have been appearing in the modeling of self organized swarm flows in social interactions and flocking dynamics, formation of networks, and queuing in supply chains. New tools from non-linear 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.