Non-local Kinetic Collisional Transport: Analysis and Numerical Methods

这项研究的总体目标是为一系列具有基础科学意义的各种自然和工程过程建立精确的数学模型和数值模拟方法，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过渡过程中非常冷气体的动力学，半导体装置中的热电子传输，用于太阳能产生氢的纳米结构的设计，与再入问题的航空动力学有关的反应分子混合物的动力学。所有这些现象都由它们的数学描述的基本结构统一起来，这些结构涉及不同类型的“粒子”相互作用。对这些数学结构的研究是这个项目的主要主题。建模和模拟将基于通过精确的晶体计算获得的数据，考虑原子修正、粗糙介质的存在等。我们将开发的相同技术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，多智能体信息传输和互联网中的社会信息动力学，仅举几例。该项目的研究目标包括一个广泛的计划，开发与统计传输方程有关的分析和数值工具，这些方程是概率应用数学的核心，统计学应用于化学、物理，在一定程度上也适用于社会动力学。它们涉及复杂相互作用系统的建模，产生与生灭动力学的马尔可夫过程相关的动力学框架。这样的统计方法导致了碰撞经典或量子Boltzmann或Smoluchowski类型的非线性积分-微分方程组。其中许多模型出现在描述纳米和介观自洽现象的短程和长程粒子相互作用模型的半经典输运理论中。来自非线性分析的新工具和新的计算策略将被开发出来，以解决长期行为、稳定性和定常模式的衰减率，以及数值解和最优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.