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CHAOS AND THERMALIZATION IN THE ONE-DIMENSIONAL
BOSE-HUBBARD MODEL IN THE CLASSICAL-FIELD APPROXIMATION
by
Amy C. Cassidy
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
May 2009
Copyright 2009 Amy C. Cassidy
Object Description
| Title | Chaos and thermalization in the one-dimensional Bose-Hubbard model in the classical-field approximation |
| Author | Cassidy, Amy C. |
| Author email | amycassi@usc.edu; amy_c_cassidy@yahoo.com |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Physics |
| School | College of Letters, Arts and Sciences |
| Date defended/completed | 2009-03-12 |
| Date submitted | 2009 |
| Restricted until | Unrestricted |
| Date published | 2009-05-08 |
| Advisor (committee chair) | Haas, Stephan |
| Advisor (committee member) |
Olshanii, Maxim Dappen, Werner Bozler, Hans Krylov, Anna |
| Abstract | One of the fundamental assertions of statistical mechanics is that the time average of a physical observable is equivalent to the average over phase space, with microcanonical measure. A system for which this is true is said to be ergodic and dynamical properties can be calculated from static phase-space averages. Dynamics of a system which is fully integrable, that is has as many conserved quantities as degrees of freedom, is constrained to a reduced phase space and thus not ergodic, although it may relax to a modified equilibrium.; In this thesis, we present a comprehensive study of chaos and thermalization of the one-dimensional Bose-Hubbard Model (BHM) within the classical field approximation. This model describes the dynamics of quantum degenerate gases in a lattice for sufficient occupation of every momentum mode and weak two-body scattering, and is of interest because of experimental advances of cooling and trapping alkali atoms in the quantum degenerate regime.; We study the chaos and its relation to thermalization. Two quantitative measures are compared: the ensemble-averaged Finite-time Maximal Lyapunov exponent, a measures of chaos and the normalized spectral entropy, a measure of the distance between the numerical time-averaged momentum distribution and the one predicted by thermodynamics. A threshold for chaos is found, which depends on two parameters, the nonlinearity and the total energy-per-particle. Below the threshold, the dynamics are regular, while far above the threshold, complete thermalization is observed, as measured by the normalized spectral entropy.; We study individual resonances in the Bose-Hubbard model to determine the criterion for chaos. The criterion based on Chirikov's method of overlapping resonances diverges in the thermodynamic limit, in contrast to the criterion parameters inferred from numerical calculations, signifying the failure of the standard Chirikov's approach.; The Ablowitz-Ladik lattice is one of several integrable models that are close to the BHM.We outline the method of Inverse Scattering Transform and generate the integrals of motion of the Ablowitz-Ladik lattice. Furthermore, we discuss the possible role of these quantities in the relaxation dynamics of the BHM. |
| Keyword | classical chaos; thermalization; one-dimensional Bose-Hubbard; Ablowitz-Ladik; discrete nonlinear Schrodinger equation; near-integrability |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m2217 |
| Contributing entity | University of Southern California |
| Rights | Cassidy, Amy C. |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Cassidy-2749 |
| Archival file | uscthesesreloadpub_Volume17/etd-Cassidy-2749.pdf |
Description
| Title | Page 1 |
| Contributing entity | University of Southern California |
| Full text | CHAOS AND THERMALIZATION IN THE ONE-DIMENSIONAL BOSE-HUBBARD MODEL IN THE CLASSICAL-FIELD APPROXIMATION by Amy C. Cassidy A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2009 Copyright 2009 Amy C. Cassidy |
