Wave Statistics in Non-Integrable Systems: From Nanostructures to Ocean Waves

不可积系统中的波浪统计：从纳米结构到海浪

• 批准号: 1205788
• 负责人: Lev Kaplan
• 金额: $ 21.59万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205788&HistoricalAwards=false
• 关键词: Wave; Statistics; Non; Integrable; Systems

This grant is for fundamental research on the statistical properties of wave functions and quantum transport in systems with a non-integrable classical limit. Experiments and applications motivating this work come from fields as diverse as current flow through two-dimensional nanostructures, rogue wave formation in the ocean, Coulomb blockade conductance in quantum dots, microwaves in irregularly-shaped electromagnetic resonators, energy transport in large structural acoustic systems, chemical reaction statistics, asymmetric optical resonators, and Casimir forces for nontrivial geometries. Interrelated research topics are: (1) Branched Flow Through Weak Correlated Random Potentials and Rogue Waves in the Ocean. Extreme event statistics are of particular interest in the context of ocean wave dynamics. The PI has obtained analytical results for the rogue wave formation probability as a function of sea parameters and will extend these techniques to include nonlinear wave evolution, finite wavelength effects, and depth variation in coastal waters, bringing closer the long-term goal of rogue wave forecasting. Similarities with other physical systems will enable improved understanding of branched flow in electron, microwave, and light scattering. (2)Chaotic Wave Functions Beyond the Random Matrix and Semiclassical Approximations: the PI has developed a robust and accurate method for extending random matrix theory predictions by systematically incorporating the non-universal short-time behavior of chaotic or diffusive systems. These techniques will be extended to general Hamiltonian systems in arbitrary dimension and to resonance wave function statistics in open systems. He will incorporate symmetry effects (including time reversal symmetry), explore the consequences of mixed classical phase space and the effects of Anderson localization. Applications include interaction matrix elements in ballistic and diffusive quantum dots, as well as energy transport in acoustic systems. (3) Vacuum Energy and Casimir Forces in Non-Integrable Geometries: The PI will investigate the vacuum self-energy in pseudointegrable and chaotic cavities in two and three dimensions. Of particular interest are the validity of the semiclassical approximations; the role of boundaries, edges, and corners; conditions under which divergences cancel between the inside and outside of a thin shell; and the relationship between the total self-energy and the local energy density. (4) Long-Time Semiclassical Accuracy: A common thread linking the above themes is the accuracy of the semiclassical approximation for long-time dynamics and eigenstates. The PI has shown previously that semiclassics at long times is more accurate in chaotic than in regular systems in two dimensions. He will apply these methods to higher-dimensional and interacting systems, and to higher-order semiclassical approximations, obtaining analytical estimates for the breakdown of the approximation in chaotic systems. Semiclassical error results will be extended to include caustics and diffraction effects. Broader impacts of the project include: Undergraduate involvement in research, with active participation of underrepresented groups, utilizing diversity-enhancement programs such as LSAMP and development of research ties with Xavier University; enhancing research opportunities for undergraduate and graduate students through direct stipend support, professional development through travel to conferences, and active participation in external collaborations; development of a new course in Chaos and Nonlinear Dynamics, targeted toward upper division undergraduate and beginning graduate students, in collaboration with faculty in mathematics and engineering; teaching of relevant introductory graduate courses in quantum and classical mechanics, with emphasis on classical quantum correspondence; and continuation of effective, highly rated teaching of physics for liberal arts majors, with a focus on modern physics and applications.

该赠款是针对具有不可整合经典限制的系统中波函数和量子传输的统计特性的基本研究。 Experiments and applications motivating this work come from fields as diverse as current flow through two-dimensional nanostructures, rogue wave formation in the ocean, Coulomb blockade conductance in quantum dots, microwaves in irregularly-shaped electromagnetic resonators, energy transport in large structural acoustic systems, chemical reaction statistics, asymmetric optical resonators, and Casimir forces for nontrivial geometries.相互关联的研究主题是：（1）通过弱相关的随机电位和海洋中的流氓波的分支流。在海浪动态的背景下，极端的事件统计尤其引起了人们的关注。 PI已获得流氓波的形成概率作为海参数的函数的分析结果，并将扩展这些技术，包括非线性波的演化，有限波长效应以及沿海水域的深度变化，从而更接近Rogue波浪预测的长期目标。与其他物理系统的相似性将使对电子，微波炉和光散射中的分支流的了解能够提高理解。 （2）混乱的波函数超出了随机矩阵和半经典近似值：PI通过系统地结合混乱或扩散系统的非通用短期行为，开发了一种强大而准确的方法来扩展随机矩阵理论预测。这些技术将扩展到任意维度的一般汉密尔顿系统，并在开放系统中共振波函数统计。他将结合对称效应（包括时间逆转对称性），探讨混合经典相空间的后果以及安德森本地化的效果。应用包括弹道和扩散量子点中的相互作用矩阵元素，以及声学系统中的能量传输。 （3）在不可融合的几何形状中真空能量和卡西米尔力：PI将在两个和三个维度中研究可构成和混沌腔中的真空自我能源。特别有趣的是半经典近似的有效性；边界，边缘和角落的作用；薄外壳内部和外部之间的分歧取消的条件；以及总自能量与局部能量密度之间的关系。 （4）长期半经典的精度：链接上述主题的一个共同线程是长期动力学和特征状态的半经典近似的精度。 PI先前已经表明，长期的半典型用途在混乱中比在二维中的常规系统中更准确。他将将这些方法应用于更高维和相互作用的系统，以及高阶的半经典近似值，从而获得混乱系统中近似值的分析估计值。半经典误差结果将扩展到包括苛性剂和衍射效应。该项目的更广泛影响包括：本科参与研究的参与以及代表性不足的群体的积极参与，利用了多样性增强计划，例如LSAMP和与Xavier University的研究关系的发展；通过直接津贴支持，通过前往会议的旅行以及积极参与外部合作来增强本科和研究生的研究机会；与数学和工程学院合作，开发了一门混乱和非线性动力学的新课程，针对上层本科生和初学者。相关的量子和古典力学研究生课程的教学，重点是经典量子通信；并继续为文科专业的有效，高度评价的物理学教学，重点是现代物理和应用。