Statistical transport of complex particle systems

该研究项目是对复杂的非保守粒子相互作用下的统计输运进行分析和数值研究。 所研究的模型包括动力学积分微分方程的玻尔兹曼型和半经典量子概率输运模型。 建模，分析和数值方法的发展将在三个主要问题领域进行：（1）与能量耗散现象有关的统计模型，包括快速颗粒流和气体混合物动力学动力学模型的硬球方程。(2)中尺度动力学系统模拟的Boltzmann-Poisson和Vlasov-Poisson系统下的电场或边界驱动力诱导的强剪切力。 (3)量子概率分布流的半经典量子轨道模型，应用于纳米级建模问题。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建模水平的联系。本研究涉及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学、社会学和物理学中，以统计学的方式描述了大量相互作用的物体的运输。 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，帮助分析这些模型及其在预测这些复杂系统行为中的应用。

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.