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.

这项研究的总体目标是在各种技术发展的前沿，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、分子混合物的反应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晶体学计算获得的数据，同时考虑到原子校正、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相关。它们包括对鸟类或鱼类等“粒子”群中的自组织流动进行建模、人口动态中的新兴共识、多智能体信息传输以及互联网中的社会信息动态等等。这些研究目标包括以下广泛的计划：与概率、统计和物理学应用数学核心的统计输运方程相关的分析和数值工具的开发。他们关注复杂相互作用系统的建模，产生与出生-死亡动力学的马尔可夫过程相关的动力学框架。这种统计方法导致了碰撞玻尔兹曼或斯莫卢科夫斯基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模型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的自洽现象。 最近，出生-死亡过程的冲突已经出现在社会互动和群体动态、网络形成和供应链排队的自组织群体流模型中。将开发非线性分析的新工具以及新的计算策略，以解决稳态模式的长期行为、稳定性和衰减率，以及数值解和最佳计算策略的定性行为。