Non-local Kinetic Collisional 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自然和工程过程开发精确的数学建模和数值模拟方法，例如聚变装置中等离子体的演化，形成玻色-爱因斯坦凝聚体的中间跃迁过程中非常冷的气体的动力学，半导体器件中的热电子传输，太阳能制氢纳米结构的设计，与再入问题的航天动力学相关的反应分子混合物动力学。所有这些现象都是由数学描述的底层结构统一起来的，其中包括不同种类的“粒子”相互作用。对这些数学结构的研究是这个项目的主要课题。建模和模拟将基于通过精确的晶体学计算获得的数据，考虑到原子修正，粗糙介质的存在等。我们将开发的同样的技术与生物和社会科学中令人兴奋的新应用有关。它们包括鸟类或鱼类等“粒子”群中自组织流动的建模，种群动态中的新兴共识，多代理信息传递和互联网中的社会信息动态，等等。该项目的研究目标包括开发与统计输运方程相关的分析和数值工具的广泛计划，这些工具是应用概率数学的核心，应用于化学、物理和一定程度上应用于社会动力学的统计。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。这样的统计方法导致碰撞经典或量子玻尔兹曼或斯摩鲁霍夫斯基型方程的非线性积分-微分系统。这些模型中有许多出现在半经典输运碰撞理论中，用于描述纳米和中尺度自洽现象的短距离和远距离粒子相互作用模型。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，稳定性和衰减率到平稳模式，以及数值解和最佳计算策略的定性行为。

The overall objective of this research is to develop accurate mathematical modeling and methods of numerical simulation for an array of diverse natural and engineering processes of fundamental scientific interest, such as evolution of plasmas in fusion devices, dynamics of very cold gases in the intermediate transition to form Bose-Einstein condensates,  hot-electron transport in semiconductor devices, design of nanostructures for the solar generation of hydrogen, dynamics of reacting molecular mixtures associated to aerospace dynamics of re-entry problems. All these phenomena are unified by the underlying structure of their mathematical description that involves "particles" interactions of different kinds. A study of these mathematical structures is the principal subject of this project. Modelling and simulation will be based on data obtained by accurate crystallographic calculations, taking into account atomistic corrections, the presence of rough media, etc. Same of the techniques that we will develop are pertinent 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. Research goals of the project 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 applied to chemistry, physics and to an extent, to 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 Smoluchowski type. Many of these models appear in the collisional theory of semi-classical transport for short- and long-range particle interactions models that describing self-consistent phenomena at nano and mesoscales.  New tools from nonlinear 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.