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.

该补助金用于对具有不可积经典极限的系统中波函数和量子输运的统计特性进行基础研究。激发这项工作的实验和应用来自不同的领域，如通过二维纳米结构的电流，海洋中的流氓波形成，量子点中的库仑阻塞电导，不规则形状的电磁谐振器中的微波，大型结构声学系统中的能量传输，化学反应统计，非对称光学谐振器，以及非平凡几何形状的卡西米尔力。相关的研究课题有：（1）通过弱相关随机势的分支流和海洋中的流氓波。极端事件统计数据在海浪动力学背景下特别令人感兴趣。PI已经获得了作为海洋参数函数的流氓波形成概率的分析结果，并将这些技术扩展到包括非线性波演变、有限波长效应和沿海沃茨的深度变化，从而更接近流氓波预报的长期目标。与其他物理系统的相似性将使我们能够更好地理解电子、微波和光散射中的分支流。（2）超越随机矩阵和半经典近似的混沌波函数：PI通过系统地结合混沌或扩散系统的非通用短时行为，开发了一种强大而准确的方法来扩展随机矩阵理论预测。这些技术将被扩展到任意维的一般哈密顿系统和开放系统中的共振波函数统计。他将结合对称性效应（包括时间反演对称性），探索混合经典相空间的后果和安德森本地化的影响。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)不可积几何中的真空能和卡西米尔力：PI将研究二维和三维中伪可积和混沌腔中的真空自能。特别感兴趣的是半经典近似的有效性;边界，边缘和角落的作用;条件下，内部和外部的薄壳之间的发散取消;和总自能和局部能量密度之间的关系。(4)长时间半经典精度：连接上述主题的一个共同线索是长时间动力学和本征态的半经典近似的精度。PI先前已经表明，在长时间内，半经典在混沌系统中比在二维规则系统中更准确。他将这些方法应用于高维和相互作用的系统，并高阶半经典近似，获得分析估计的崩溃近似在混沌系统。半经典误差结果将被扩展到包括焦散和衍射效应。该项目的更广泛的影响包括：本科生参与研究，积极参与代表性不足的群体，利用多样性增强方案，如LSAMP和与泽维尔大学的研究关系的发展;通过直接的津贴支持，通过旅行到会议的专业发展，并积极参与外部合作，提高本科生和研究生的研究机会;在混沌和非线性动力学的新课程的发展，针对高年级本科生和开始研究生，与数学和工程学院合作;量子和经典力学的相关介绍性研究生课程的教学，重点是经典量子对应;继续为文科专业提供有效的、高评价的物理教学，重点是现代物理学及其应用。