CAREER: Transport Phenomena in Quantum Systems and Random Media

这一职业奖支持首席研究员在“交通现象”一般领域的研究和教育计划。输运现象在自然界中无处不在，无论是在我们宏观世界的大尺度上，还是在微观粒子的非常小的尺度上。它们描述了系统之间的交互和交换，以及如何将大量的兴趣从一个位置传输到另一个位置。该项目侧重于这些现象的两个重要实例：复杂环境中的光波或声波的传输，以及纳米级设备中的粒子传输。这两个主题在电信、无损检测、医学成像和纳米电子学等各个基本领域都有许多应用。通常情况下，输运现象是多尺度的，从这个意义上讲，潜在的物理涉及到各种不同量级的空间或时间尺度。因此，运输过程的描述和计算机模拟极其困难，这就是数学分析发挥核心作用的地方。通过识别一组重要参数，可以将原始模型简化为更易于分析和计算的简化模型。这种渐近分析的概念是该项目两个主题的共同主线。该项目将提出新的数学模型和技术，以及复杂介质和量子系统中传输现象的新计算方法。这项研究将特别影响无序介质中目标或缺陷局部化的领域，以及半导体的建模。该项目将研究与教育计划相结合，旨在覆盖从高中到博士生的不同世代。该项目涉及与交通现象相关的两个主题。第一个涉及纳米器件的量子流体动力学输运模型，该模型在量子尺度上提供了对动力学的简化而准确的宏观描述。主要目标是为最近的正式模型建立完整的数学理论，为它们的模拟开发数值方法，并为未来的应用验证模型。该理论的一个关键组成部分是“量子矩问题”，它是经典的测量矩问题的密度算子的推广。第二个主题涉及随机介质中的逆问题。主要目标是开发变革性的基于运输的方法，用于在强扩散环境中成像，目前还没有有效的技术可用。这些方法是基于对信息从微观尺度到宏观尺度的传递的理解。重点将放在具有长期相关性的随机介质上，例如湍流大气，这为项目中提出的方法提供了最终的基准。使用优化技术的快速强度成像算法的问题也将被解决。

This CAREER award supports the research and education program of the Principal Investigator in the general area of "transport phenomena." Transport phenomena are ubiquitous in nature, whether at the large scale of our macroscopic world or at the very small scale of microscopic particles. They describe interactions and exchanges between systems, and how quantities of interest are transported from one location to another. This project focuses on two important instances of these phenomena: the transport of light or sound waves in complex environments, and the transport of particles in devices at the nanometer scale. These two topics have many applications in various fundamental domains such as telecommunications, non-destructive testing, medical imaging, and nanoelectronics. Very often, transport phenomena are multiscale, in the sense that the underlying physics involves various spatial or temporal scales of very different orders of magnitude. As a consequence, the description and the computer simulation of transport processes are extremely difficult, and this is where mathematical analysis plays a central role. By identifying a set of important parameters, the original models can be simplified into reduced models, more amenable to analysis and computations. This notion of asymptotic analysis is the common thread of the two topics of the project. The project will advance new mathematical models and techniques, as well as new computational methods for transport phenomena in complex media and in quantum systems. The research will impact, in particular, the field of target or defect localization in disordered media, as well as the modeling of semiconductors. The project integrates research with an educational plan that is designed to reach various generations, from high school to doctoral students.This project addresses two topics related to transport phenomena. The first one concerns quantum hydrodynamical transport models for nanodevices, which offer a reduced yet accurate macroscopic description of the dynamics at the quantum scale. The main objectives are to build a complete mathematical theory for recent formal models, to develop numerical methods for their simulations, and to validate the models for future applications. A key ingredient in the theory is the "quantum moment problem," which is a generalization to density operators of the classical moment problem for measures. The second topic concerns inverse problems in random media. The main goal is to develop transformative transport-based methods for imaging in strongly diffusive environments for which no efficient techniques are available as of today. Such methods are based upon understanding the transfer of information from the microscopic scale to the macroscopic scale. The stress will be put on random media with long-range correlations, such as the turbulent atmosphere, which offer an ultimate benchmark for the methods proposed in the project. The problem of fast intensity imaging algorithms using optimization techniques will also be addressed.