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，我们对一类重要的系统特别感兴趣，即低维纳米结构（ldn）。它们可以自我组装，而且它们的非凡特性可以被设计用于手边的特定应用。因此，它们的潜力几乎是无限的。由于许多在更大尺度上可以忽略不计的影响，对于ldn来说不能再被忽视了，因此数学建模正在决定性地成为他们研究的主要工具。****提出的研究计划旨在进一步推进用于分析ldn和其他感兴趣系统的多尺度数学模型的开发和应用，并将其扩展到分析此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这些影响必不可少的系统，系统地解释几何相位和时间相关耦合效应。自首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已成功地应用于广泛的经典和混合量子连续统系统。然而，这一领域的许多数学挑战仍处于几乎未被探索的阶段。该项目将充分利用PI小组已经取得的专业知识和进步。应用方面，重点将放在ldn和分子和纳米水平的几类生物系统上，特别是核糖核酸纳米结构和光合复合体，以及混合量子连续统系统。****该程序将首先允许系统地研究这些系统的性质，其中几何相位和时间相关耦合是必不可少的。尽管这些系统在自然和人为环境中普遍存在，但目前还缺乏对几类重要问题的系统研究。其次，它将更好地理解几何相诱导力与重要动力学现象之间的联系，这是一个具有重大基础和技术兴趣的领域。第三，鉴于其普遍存在的性质，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问题及其应用。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类相当普遍的问题，其中几何相位和动态耦合效应对系统特性的影响是显著的。这些结果可能不仅仅局限于纳米结构，而且可以用于科学和工程中其他重要系统的研究

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.**