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.
