Statistical transport of complex particle systems

本研究项目是对复杂非保守粒子相互作用下的统计输运进行分析和数值研究。研究的模型包括玻尔兹曼型动力学积分微分方程和半经典量子概率输运模型。模拟、分析和发展数值方法将在三个主要问题领域进行：(1)与能量耗散现象相关的统计模型，包括用于快速颗粒流动和气体混合物动力学模型的硬球方程。(2)由电场或边界驱动力诱导的强剪切力作用下的Boltzmann-Poisson和Vlasov-Poisson系统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迹模型及其在纳米尺度建模问题中的应用。将开发非线性分析的新工具以及新的计算策略，以建立从量子到动力学到流体水平的建模联系。这项研究涉及许多重要科学问题的数学模型的发展和分析。这些模型出现在生物、社会和物理科学中，用统计学的方法描述了大量相互作用的物体的传输。应用包括快速颗粒流的建模，化学反应气体混合物，经济系统中的财富分配，意见动力学和半导体行为。该项目将开发新的数学工具，以帮助分析这些模型并将其用于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.