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，继续给我们的日常生活带来巨大的革命性变化。在这个项目中，我们特别感兴趣的是一类重要的此类系统，即低维纳米结构(LDN)。它们可以自我组装，其非凡的性能可以根据手头的特定应用进行设计。因此，它们的潜力几乎是无限的。由于在较大范围内可以忽略的许多影响对发展网络来说不能再被忽视，因此数学模型正在果断地成为其研究的一个主要工具。 拟议的研究方案旨在进一步发展和应用多尺度数学模型来分析LDN和其他感兴趣的系统，并将其扩展到分析这类系统的新的最先进的数学和计算框架。具体地说，主要目标是通过发展数学上连贯的方法来研究那些对几何阶段和依赖时间的耦合效应至关重要的系统，从而系统地解释这些影响。自从它第一次出现在量子系统中以来，几何相位这一优雅简单的数学概念也成功地应用于各种经典的和混合的量子连续谱系统。然而，这一领域的许多数学挑战仍处于鲜为人知的地平线上。 该计划将利用PI小组发展的专业知识和已经取得的进展。在应用方面，将主要关注低密度脂蛋白以及分子和纳米尺度上的几类生物系统，特别是核糖核酸纳米结构和光合作用复合体，以及混合量子-连续体系。 首先，该程序将允许系统地研究这类系统的性质，其中几何相位和依赖时间的耦合是必不可少的。尽管这样的系统在自然和人工环境中普遍存在，但目前还缺乏对几类重要问题的系统研究。其次，它将提供一个更好的理解几何相诱导力和在一个具有重大基础和技术意义的领域中的重要动力学现象之间的联系。第三，鉴于它们无处不在的性质，预计本提案中开发的模型和工具将有助于解决其他具有挑战性的数学及其应用问题。事实上，对于几何相位和动态耦合效应对系统性质的影响很大的一类非常一般的问题，预计在该程序内开发的方法和工具是必不可少的。这些结果可能不仅限于纳米结构，还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.