The objective of this proposal focus on a broad program in analytical and numerical problems associated with the complex nature of non-conservative particle interactions in a kinetic (integro-differential Boltzmann type) framework of statistical transport equations. These models appear and the studies of semi-classical transport for short and long range interactions models that describe self-consistent phenomena at nano and mesoscales. More recently they have been appearing in the modeling of social interactions and the formation of networks such as Internet social dynamics and queuing in supply chains. New tools from non-linear analysis as well as new computational strategies need to be developed in order to address the linking from quantum to kinetic to fluid level of modeling. Of special interest is the understanding and developing of long time behaviour of the model, decay rates to stable modes, qualitative behavior of the solutions and optimal computational strategy.
This area of research, essentially Applied Mathematics and Probability and Statistics, is related to Mathematical and Statistical Physics with remarkable new applications to non-linear dynamics modeling in Bio Sciences and Social Sciences as well. The investigated problems range into a broad area of statistical transport from modeling and prediction phenomena of rapid granular flows, reacting gas molecule mixtures, transport modeling at atomistic scale, particle swarms, opinion dynamics, multi-agent information transfer flow, and social information dynamics in Internet to name a few.