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

本研究关注不同尺度下自洽带电粒子输运系统理论中的数学问题，特别关注强强迫（由于流动区域的不均匀性导致的高内建电场）下动力学输运中的数学问题，以及在这些条件下推导的各向异性流体动力学模型的层次。这些模型自然出现时，在异质结构，如半导体器件的非平衡电流流动的研究。类似的模型用于描述离子通道中的生物运输，这是由于通过细胞膜的显著电活性。 计划研究的新方面之一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动力学和宏观水平连接传输模型的尺度。在这两种情况下，分析研究是由相应的模型方程的解决方案的数值模拟来补充。 本研究致力于描述半导体和生物膜中电子和离子传输的数学模型，其中一个主要挑战来自于这样一个事实，即在这些现象中，发生在非常不同的长度和时间尺度上的过程是相互耦合和影响的。 数学工作将导致更快的计算方法来模拟这种现象，同时更准确。像微电子学和药理学这样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.