Collisional Kinetic Transport: Analysis and Numerical Methods

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精确的分析建模和仿真，这些现象处于各种技术发展的边缘，如聚变模型中的等离子体演化，形成玻色-爱因斯坦凝聚的中间跃迁中非常冷的气体的建模，半导体器件中的热电子输运，用于太阳能发电的纳米结构，以及与航天动力学相关的反应和多原子分子混合物，比如那些重返大气层的问题。建模和模拟将基于精确的晶体学计算获得的数据，考虑到原子修正和粗糙介质的存在。将开发的一些技术也与生物社会科学中非线性动力学建模的令人兴奋的新应用相关，例如自组织流的建模，其中“粒子”群，如鸟或鱼，与流体动力学耦合，种群动力学中的新兴共识，多代理信息传递和互联网中的社会信息动力学，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，为他们进入学术界、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项目，开发与统计输运方程和系统相关的分析和数值工具，这些工具是应用数学的核心，应用于化学、物理以及生物和社会动力学的统计。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。这种统计方法导致碰撞经典或量子玻尔兹曼或斯莫鲁科夫斯基型方程的非线性积分-微分系统。计算方案将充分设计和分析，以确保一致性，稳定性，误差估计控制，和收敛速度平衡。这些模型中的许多出现在半经典输运碰撞理论中，用于描述纳米和中观尺度上自洽现象的短期和长期粒子相互作用模型。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、稳定性和平稳模式的衰减率，以及数值解和最佳计算策略的定性行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.