Statistical transport of complex particle systems

本研究项目是复杂的非保守粒子相互作用下统计输运的解析和数值研究。所研究的模型包括Boltzmann型动力学积分-微分方程组和半经典量子几率输运模型。将在三个主要问题领域进行建模、分析和数值方法的发展：(1)与能量耗散现象有关的统计模型，包括用于快速颗粒流动力学模型和气体混合物动力学模型的硬球方程。(2)由Boltzmann-Poisson系统和Vlasov-Poisson系统在电场或边界驱动力诱导的强剪切力作用下模拟的中尺度动力学系统。(3)量子概率分布流动的半经典量子轨道模型及其在纳米尺度建模中的应用。非线性分析的新工具以及新的计算策略将被开发出来，以建立从量子到动力学再到流体的建模层次。这项研究涉及到一些重要科学问题的数学模型的开发和分析。这些模型出现在生物、社会和物理科学中，从统计学上描述了大量相互作用的物体的运输。应用包括对快速颗粒流、化学反应气体混合物、经济系统中的财富分配、舆论动力学和半导体行为的建模。这个项目将开发新的数学工具，帮助分析这些模型，并使用它们来预测这些复杂系统的行为。

This research project is an analytical and numerical study of statistical transport under complex, non-conservative particle interactions. The models under study include kinetic integro-differential equations of Boltzmann type and semiclassical quantum probability transport models. Modeling, analysis, and development of numerical methods will be conducted in three primary problem areas:(1) Statistical models related to energy-dissipative phenomena, including hard sphere equations for kinetic models of rapid granular flow and dynamics of gas mixtures. (2) Mesoscale kinetic systems modeled by Boltzmann-Poisson and Vlasov-Poisson systems under strong shear forces induced by electric fields or by boundary driving forces. (3) Semiclassical quantum trajectory models for flow of quantum probability distributions, with application to problems in nanoscale modeling.New tools from non-linear analysis as well as new computational strategies will be developed to establish links from quantum to kinetic to fluid levels of modeling.This research concerns the development and analysis of mathematical models for a number of important scientific problems. The models, which arise in biological, social, and physical sciences, describe statistically the transport of large numbers of interacting objects. Applications include modeling of rapid granular flows, chemically reacting gas mixtures, distribution of wealth in economic systems, opinion dynamics, and semiconductor behavior. This project will develop new mathematical tools that aid in the analysis of such models and their use in predicting the behavior of these complicated systems.