Mathematical Issues in the Modeling and Simulation of Self-Consistent Charge Particle Transport

这项研究涉及不同尺度下自洽带电粒子输运系统理论中的数学问题，特别是解决强强迫(由于流区的不均匀而产生的高内建电场)下的运动输运的数学问题，以及在这些条件下得到的各向异性流体动力学模型。这些模型是在研究半导体器件等非均质结构中的非平衡电流流动时自然产生的。类似的模型也被用来描述离子通道中的生物传输，这是由于通过细胞膜的显著电活动。计划中的研究的一个新方面是对过渡层的数学和计算理解，其中涉及在低电场和高电场效应下，通过传输模型的动力学和宏观水平将尺度联系起来。在这两种情况下，分析研究将得到相应模型方程解的数值模拟的补充。这项研究解决了用于描述半导体和生物膜中电子和离子传输的数学模型。主要挑战之一来自于这样一个事实，即在这些现象中，发生在非常不同长度和时间尺度上的过程是相互耦合和相互影响的。数学工作将导致模拟这类现象的更快的计算方法，同时也更准确。微电子学和药理学等不同的技术领域最终将从这项工作中受益。

This research concerns mathematical problems in the theory of self-consistent charged-particle transport systems at different scales.Of particular interest is to address the mathematical issues in kinetictransport under strong forcing (a high built-in electric field due to inhomogeneities of the flow region), and the hierarchy of anisotropic hydrodynamic models derived under these conditions. These models arise naturally when non-equilibrium current flow in heterogeneous structures such as semiconductor devices is studied. Similar models are used to describe biological transport in ionic channels, due to significant electrical activity through cellular membranes. One of the new aspects of the planned research is the mathematical and computational understanding of transition layers that involve the linking of scales through kinetic and macroscopic levels for transport models, both under low and high electric field effects. In both cases, analytical studies are to be complemented by numerical simulationof solutions of the corresponding model equations. This research addresses mathematical models that are used to describeelectron and ion transport in semiconductors and in biological membranes.One of the main challenges comes from the fact that in these phenomena, processes that occur at very different length and time scale are coupled and influence each other. The mathematical work will lead to faster computational methods for the simulation of such phenomena which are at the same time more accurate. Technological areas which are as different as microelectronics and pharmacology will ultimately benefit from this work.