The Kinetics of Interacting Particle Systems: Theory and Numerical Methods

这项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、用于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的等离子体演化，对一系列具有基本科学意义的不同现象进行准确的建模和模拟。建模和模拟将基于通过精确的结晶学计算获得的数据，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。这些研究目标包括开发与统计传输方程相关的分析和数值工具的广泛计划，这些工具是概率、统计学和物理学应用数学的核心。它们涉及复杂相互作用系统的建模，产生与生灭动力学的马尔可夫过程相关的动力学框架。这样的统计方法导致了碰撞Boltzmann或Smolukowski类型的非线性积分-微分方程组。其中许多模型出现在短程和长程相互作用的半经典输运碰撞理论中，这些模型描述了纳米和介观尺度上的自洽现象。最近，生灭过程的碰撞已经出现在社会互动和集群动力学中的自组织群流的建模、网络的形成和供应链中的排队中。将开发来自非线性分析的新工具以及新的计算策略，以解决长时间行为、稳定性和衰减率到平稳模式，以及数值解的定性行为和最优计算策略。

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.