The Kinetics of Interacting Particle Systems: Theory and Numerical Methods

本研究的总体目标是在各种技术发展的边缘，如半导体器件中的热电子传输，太阳能产生氢的纳米结构，反应分子混合物以及聚变模型中等离子体的演化，为一系列具有基础科学兴趣的不同现象开发精确的建模和模拟。建模和模拟将基于通过精确的晶体学计算获得的数据，考虑到原子修正，粗糙介质的存在等。许多有待开发的技术也与生物和社会科学中令人兴奋的新应用有关。它们包括鸟类或鱼类等“粒子”群中自组织流动的建模，种群动态中的新兴共识，多代理信息传递和互联网中的社会信息动态，等等。这些研究目标包括与统计输运方程相关的分析和数值工具的发展的广泛计划，这些工具是应用数学在概率，统计和物理中的核心。它们涉及复杂相互作用系统的建模，产生与马尔可夫生-死动力学过程相关的动力学框架。这样的统计方法导致了碰撞玻尔兹曼或斯摩可夫斯基型非线性积分-微分方程组。这些模型中的许多出现在半经典输运碰撞理论中，用于描述纳米和中尺度自洽现象的短程和长距离相互作用模型。最近，出生-死亡过程的碰撞已经出现在社会互动和群集动力学、网络形成和供应链排队的自组织群体流的建模中。来自非线性分析的新工具以及新的计算策略将被开发出来，以解决长期行为，稳定性和衰减率到平稳模式，以及数值解和最佳计算策略的定性行为。

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.