Doctoral Dissertations

Xiazhou Yang

Date of Award

8-1993

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Physics

Major Professor

Joachim Burgdörfer

Committee Members

Edward Harris, Tom Callcott, John Quinn, John Bloor

Abstract

We present theoretical studies of the dynamical origin for the existence of universality classes in quantum spectral fluctuations. The evolution of the quantum spectrum as a function of a control parameter governing the strength of a nonintegrable perturbation can be mapped onto the molecular dynamics of a many-body system. The perturbation causes chaotic motion in the classical dynamics. We show that the classical transition from predominantly regular motion to global chaos is accompanied by a transition from a nonequilibrium to an equilibrium state in the parametric motion of quantum energy levels. The discussion of the source of ergodicity in the parametric motion leads to an equilibrium distribution function which provides an explanation for the existence of the universal fluctuations in the quantum spectra of classically chaotic systems. The microscopic mechanism for the transition from a nonequilibrium to an equilibrium state is the existence of avoided crossings. The width distribution of avoided crossings in quantum systems whose classical counterparts possess a mixed phase space of regular and chaotic dynamics is shown to consist of two components: a near Gaussian distribution suggested by random matrix theory for the chaotic component and an approximately 6-shaped component originating from tori in the regular portion of phase space. The relative weight of these two components is closely related to the structure of classical phase space. The study of relaxation processes in the parametric motion of energy levels reveals that the parametric correlation functions are sensitively dependent on the underlying classical dynamics, and the transition from classically regular to chaotic motion manifests itself in the disappearance of parametric diffusion in quantum spectrum. The parametric motion of resonances is discussed. The two-body approximation indicates that the existence of the repulsion between resonances is independent of the underlying classical dynamics. We suggest that the transition to the thermal equilibrium state in the parametric motion of resonances is the signature of classical chaos.

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