Statistics of Quantum Energy Levels of Integrable Systems and a Stochastic Network Model with Applications to Natural and Social Sciences

I have done two projects in my PhD period: first, the statistics of quantum energy levels of integrable systems and second, a stochastic network model with applications to natural and social sciences. Quantum chaos studies level statistics and wave-function characteristics in the semiclasscal regime where the system action is several orders of magnitude larger than Planck’s constant. The level statistics of classically integrable systems and classically chaotic ones are different. The levels of classically chaotic systems repel each other while these of classically integrable ones seem to be uncorrelated. We have studied level statistics in semiclassical spectrum of classically integrable systems without extra degeneracies. Our new findings are as follows. We developed a parametric averaging method to achieve ensemble averaging. We found the interval level number variance displays persistent oscillations. Contrary to previous belief, the nearest-neighbor spacing distribution of integrable systems displays some repulsion. The theoretical explanations of level statistics are based on both semiclassical theory and quantum mechanical derivation. Distributions of mass of species in the field of biology, human response times in the field of psychology, personal wealth distribution in the field of economics, and stock return in finance, all display power-law tails. It is theoretically and practically meaningful to understand the characteristics and origin of these power-law tails. We study a stochastic network model that is able to generate generalized inverse gamma distribution with powerlaw tail. We apply this distribution to the field of psychology and find good fit of data of with human response times. At last, we extend the model to study the stock return. UNIVERSITY OF CINCINNATI Date: May 13, 2013 Author: Tao Ma Title: Statistics of Quantum Energy Levels of Integrable Systems and a Stochastic Network Model with Applications to Natural and Social Sciences Department: Physics Degree: Ph.D. Convocation: May 13 Year: 2013 Permission is herewith granted to University of Cincinnati to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, AND NEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’S WRITTEN PERMISSION. THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINED FOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THIS THESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPER ACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USE IS CLEARLY ACKNOWLEDGED. Acknowledgements I thank the Department of Physics of University of Cincinati for supporting me financially in my PhD study. I thank Rostislav A. Serota, my advisor, for his many suggestions and kind support during my PhD study. When I first came to UC I do not know much about how to do research and do not have many skills. Thanks to his advising in the last five years, I know how to approach and solve problems and also have research skills now. I thank John G. Holden for his collaboration in the project of stochastic network model. I thank University of Cincinati for giving me the Distinguished Dissertation Completion Fellowship and thank Rostislav A. Serota, Michael Ma, John G. Holden, and L.C. R. Wijewardhana for their help in my application for the fellowship. Part of the dissertation is done under the support of the fellowship. Part of the computation in the dissertation is done on a computer system of Ohio Supercomputer Center under grant No. PES0340 and Michael Sokoloff’s workstation. Dedicated to my Parents Guanqing Ma and Ying Yao.