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

该基金用于对具有不可积经典极限的系统中的波函数和量子输运的统计性质进行基础研究。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、海洋中的异常波形成、量子点中的库仑阻塞电导、不规则形状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、不对称光学谐振器以及非平凡几何的卡西米尔力。相关的研究课题有：(1)海洋弱相关随机势的分支流动与异常浪。在海浪动力学的背景下，极端事件统计具有特别的意义。PI已经获得了异常浪形成概率作为海洋参数函数的分析结果，并将扩展这些技术，包括非线性波浪演变，有限波长效应和沿海水域的深度变化，使异常浪预报的长期目标更加接近。与其他物理系统的相似之处将使人们更好地理解电子、微波和光散射中的支流。(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普遍短时行为，开发了一种鲁棒和精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。他将结合对称效应（包括时间反转对称），探索混合经典相空间的后果和安德森局域化的影响。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将研究二维和三维伪可积和混沌腔中的真空自能。特别令人感兴趣的是半经典近似的有效性；边界、边和角的作用；薄壳内外的散度相互抵消的条件；以及总自能与局部能量密度的关系。(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特征态的半经典近似的精度。PI先前已经表明，长时间的半经典在二维混沌系统中比在规则系统中更精确。他将把这些方法应用于高维和相互作用系统，以及高阶半经典近似，获得混沌系统中近似分解的分析估计。半经典误差结果将扩展到包括焦散和衍射效应。该项目的更广泛影响包括：本科生参与研究，与代表性不足的群体积极参与，利用多样性增强计划，如LSAMP和发展与泽维尔大学的研究联系；增加本科生和研究生的研究机会，包括直接津贴、参加会议的专业发展，以及积极参与外部合作；与数学和工程学院合作，为高年级本科生和研究生开设混沌与非线性动力学新课程；量子与经典力学研究生相关入门课程的教学，重点是经典量子对应；继续为文科专业学生提供高效、高评价的物理教学，重点是现代物理及其应用。

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.