Collisional Kinetic Transport: Analysis and Numerical Methods

这项研究的总体目标是为一系列具有根本科学意义的各种现象开发准确的分析建模和模拟，这些现象处于各种技术发展的边缘，例如聚变模型中的等离子体演化，中间过渡中非常冷的气体的建模以形成玻色-爱因斯坦凝聚，半导体器件中的热电子传输，太阳能制氢的纳米结构，以及与航天动力学有关的反应和多原子分子混合物，例如再入问题。建模和模拟将基于通过精确的晶体学计算获得的数据，考虑原子校正和粗糙介质的存在。一些将被开发的技术也是相关的令人兴奋的新的应用程序在生物社会科学的非线性动力学建模，如自组织流的“粒子”群，如鸟类或鱼类，耦合到流体动力学，新兴的共识，人口动态，多代理信息传输和社会信息动态在互联网上的建模，仅举几例。最重要的是，该项目为研究生提供了研究培训机会，使他们为从学术界到国家实验室和工业界的就业市场做好准备。这些研究目标包括一个广泛的计划，开发与概率统计输运方程和系统相关的分析和数值工具，应用于化学的统计，物理学以及生物学和社会动力学。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。这样的统计方法导致碰撞经典或量子玻尔兹曼或Smolukowski型方程的非线性积分微分系统。计算方案将被充分设计和分析，以确保一致性，稳定性，误差估计控制和收敛速度平衡。许多这些模型出现在碰撞理论的半经典运输的短期和长期的粒子相互作用模型，描述自洽现象在纳米和介观尺度。 将开发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。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The overall objective of this research is to develop accurate analytical modeling and simulation for a series of diverse phenomena of fundamental scientific interest, at the edge of various technological developments such as plasma evolution in fusion models, modeling of very cold gases in the intermediate transition to form Bose-Einstein condensation, hot-electron transport in semiconductor devices, nanostructures for solar generation of hydrogen, and reacting and polyatomic molecular mixtures associated with aerospace dynamics such as those in re-entry problems. Modeling and simulation will be based on data obtained by accurate crystallographic calculations, considering atomistic corrections, and the presence of rough media. Some of the techniques that will be developed are also pertinent to exciting new applications to non-linear dynamics modeling in bio-social sciences, such as modeling of self-organized flows where "particle" swarms, like birds or fish, couple to fluid dynamics, emerging consensus in population dynamics, multi-agent information transfer and social information dynamics in Internet, to name a few. Most significantly, this project provides research training opportunities for graduate students that prepare them to a job market that ranges from academia, to national labs, and industry.These research goals comprise a broad program in the development of analytical and numerical tools associated with statistical transport equations and systems at the core of applied mathematics in probability, statistics applied to chemistry, physics as well as to biological and social dynamics as well. 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 classical or quantum Boltzmann or Smolukowski type. Computational schemes will be fully designed and analyzed to secure consistency, stability, error estimates control, and convergence rates to equilibrium. Many of these models appear in the collisional theory of semi-classical transport for short- and long-range particle interactions models that describe self-consistent phenomena at nano and meso-scales. 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.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.