The Kinetics of Interacting Particle Systems: Theory and Numerical Methods

相互作用粒子系统的动力学：理论和数值方法

• 批准号: 1413064
• 负责人: Irene Gamba
• 金额: $ 24万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-15 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1413064&HistoricalAwards=false
• 关键词: Kinetics; Interacting; Particle; Systems; Theory

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.

本研究的总体目标是为一系列具有基础科学兴趣的不同现象开发准确的建模和模拟，这些现象处于各种技术发展的边缘，例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。建模和模拟将基于通过精确的晶体学计算获得的数据，考虑到原子校正，粗糙介质的存在等，许多技术的开发也是相关的，令人兴奋的新应用在生物和社会科学。这些研究领域包括鸟或鱼等“粒子”群中自组织流动的建模、种群动力学中的新兴共识、多智能体信息传递以及互联网中的社会信息动力学等。这些研究目标包括一个广泛的计划，即开发与概率论、统计学和物理学中应用数学核心的统计输运方程相关的分析和数值工具。它们涉及复杂的相互作用系统的建模，产生与马尔可夫过程的生灭动力学相关的动力学框架。这种统计方法导致碰撞玻尔兹曼或Smolukowski型方程的非线性积分微分系统。这些模型中的许多出现在描述纳米和介观尺度自洽现象的短程和长程相互作用模型的半经典输运碰撞理论中。 最近，出生-死亡过程的碰撞已经出现在社会相互作用和群集动力学中的自组织群体流、网络的形成和供应链中的排队的建模中。将开发来自非线性分析的新工具以及新的计算策略，以解决长期行为、稳定性和稳态模式的衰减率，以及数值解的定性行为和最佳计算策略。