Kinetic transport and dynamics in complex interacting systems: analysis and simulations

本提案的目标集中在与统计输运方程的动力学（积分-微分玻尔兹曼型）框架中非保守粒子相互作用的复杂性相关的分析和数值问题的广泛程序。这些模式的出现以及对描述纳米和中尺度自洽现象的短程和长距离相互作用模式的半经典输运的研究。最近，它们出现在社会互动和网络形成的建模中，如互联网社会动态和供应链排队。需要开发非线性分析的新工具以及新的计算策略，以解决从量子到动力学到流体水平建模的联系。特别感兴趣的是对模型长期行为的理解和发展，对稳定模式的衰减率，解的定性行为和最优计算策略。这一研究领域，本质上是应用数学和概率与统计，与数学和统计物理相关，在生物科学和社会科学的非线性动力学建模中也有显著的新应用。所研究的问题涵盖了统计传输的广泛领域，从快速颗粒流动的建模和预测现象，反应气体分子混合物，原子尺度的传输建模，粒子群，意见动态，多智能体信息传输流，以及互联网中的社会信息动态等等。

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.