Multiscale models for nanostructures with geometric phases and time-dependent coupling

微小的系统可以小到人类头发直径的千分之一，继续为我们的日常生活带来巨大的革命性变化。在这个计划中，我们特别感兴趣的一类重要的系统称为低维纳米结构（LDN）。它们可以自组装，其非凡的特性可以被设计成手头的特定应用。因此，它们的潜力几乎是无限的。由于许多在较大尺度上可以忽略不计的影响，对于LDN来说再也不能忽视了，因此数学建模决定性地成为他们研究的主要工具。 拟议的研究计划旨在进一步推进多尺度数学模型的开发和应用，以分析LDN和其他感兴趣的系统，并将其扩展到一个新的最先进的数学和计算框架，用于分析此类系统。具体而言，主要目标是系统地考虑几何相位和时间依赖性耦合效应，通过发展数学上连贯的方法来研究这些效应至关重要的系统。自从它第一次出现在量子系统中以来，几何相位的优雅简单的数学概念也被成功地应用于广泛的经典和混合量子连续系统。然而，在这一领域的许多数学挑战仍然只是很少探索的视野。 该计划将利用PI小组已经取得的专业知识和进步。在应用方面，主要关注LDN和分子和纳米级的几类生物系统，特别是核糖核酸纳米结构和光合复合物，以及混合量子连续系统。 该计划将，首先，允许这样的系统的几何相位和时间相关的耦合是必不可少的属性的系统研究。虽然这样的系统是普遍存在于自然和人为环境中，他们的系统研究的几个重要类别的问题目前还没有。其次，它将提供一个更好的理解之间的连接的几何相位诱导力和重要的动力学现象，在一个领域的巨大的基础和技术利益。第三，鉴于其普遍存在的性质，预计本提案中开发的模型和工具将有助于解决数学及其应用的其他挑战性问题。事实上，在这个程序中开发的方法和工具，预计将是不可缺少的一个相当一般的一类问题的几何相位和动态耦合效应对系统的性能的影响是显着的。这些结果可能不仅限于纳米结构，而且可以用于研究科学和工程中的其他重要系统。

Tiny systems that can be as small as 1/1000th the diameter of a human hair continue to bring tremendous revolutionary changes in our everyday lives. In this program, we are particularly interested in an important class of such systems known as low dimensional nanostructures (LDNs). They can be self-assembled and their extraordinary properties can be engineered to a specific application at hand. As a result, their potential is virtually unlimited. Since many effects, that were negligible at larger scales, cannot be ignored any longer for LDNs, the mathematical modelling is decisively becoming a major tool in their studies. The proposed research program is aimed at further advancement in the development and applications of multiscale mathematical models for the analysis of LDNs and other systems of interest, and expanding it to a new state-of-the-art mathematical and computational framework for analyzing such systems. Specifically, the main goal is to account systematically for geometric phases and time-dependent coupling effects by developing mathematically coherent approaches to the study of the systems where such effects are essential. Since its first appearance in quantum systems, the elegantly simple mathematical concept of geometric phase has also been successfully applied to a wide range of classical and hybrid quantum-continuum systems. Nevertheless, many mathematical challenges in this field are still on only scarcely explored horizons. The program will capitalize on the developed expertise and advances already made by the PI’s group. Application-wise, major focus will be given to LDNs and to several classes of biosystems at the molecular and nanoscale levels, in particular Ribonucleic acid nanostructures and photosynthetic complexes, as well as to hybrid quantum-continuum systems. The program will, firstly, allow a systematic study of properties of such systems where geometric phases and time-dependent couplings are essential. Although such systems are pervasive in natural and man-made environments, their systematic studies for several important classes of problems are currently absent. Secondly, it will provide a better understanding of the connection between the geometric-phase-induced forces and important dynamic phenomena in a field of great fundamental and technological interest. Thirdly, given their ubiquitous nature, it is expected that the models and tools developed in this proposal will assist in addressing other challenging problems of mathematics and its applications. Indeed, methods and tools to be developed within this program are expected to be indispensable for a quite general class of problems where the influence of geometric phases and dynamic coupling effects on the properties of the systems is significant. The results may not be restricted to just the nanostructures and can be useful in studying other important systems in science and engineering.