Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 98-102). Available also in a digital version from Dissertation Abstracts.

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Orientador: Gastão Inácio Krein / Banca: Iberê Luiz Caldas / Banca: Felipe Barbedo Rizzato / Resumo: Uma das características mais importantes das teorias de gauge não-Abelianas é a não-linearidade das equações de campo clássicas. Mostra-se no contexto da teoria de Yang-Mills que essa característica pode fazer com que o campo de gauge apresente comportamento caótico. Isso pode acontecer mesmo quando estivermos considerando a dinâmica do campo na ausência de fontes, isto é, o vácuo da teoria de Yang-Mills. Discutimos a relação entre os comportamentos caótico e ergódico. Em seguida, introduzimos a formulação de Berdichevsky da Mecânica Estatística Clássica para sistemas dinâmicos Hamiltonianos que são ergódicos e possuem poucos graus de liberdade. A Mecânica Estatística de Berdichevsky é usada para estudar a situação mais simples numa teoria de gauge não-Abeliana onde as variáveis de campo são caóticas e o espaço de fase correspondente tem a propriedade geométrica necessária. Mostramos que, para os propósitos desse estudo, um par de campos escalares complexos deve ser incluído no problema. Mais precisamente, analisamos o modelo de Higgs não-Abeliano; a Lagrangiana da teoria considerada possui uma simetria SU(2). A transição de uma descrição dinâmica do sistema de YangMills-Higgs (fora do equilíbrio termodinâmico) para uma descrição termodinâmica (quando ele atingiu o equilíbrio) é investigada numericamente. Mostra-se que depois de um tempo suficientemente longo as soluções numéricas se comportam de tal maneira que o sistema pode ser descrito de um jeito mais simples através de grandezas como a temperatura, calculadas de acordo com as prescriçõees da Mecânica Estatística de equilíbrio. Estas são previstas analiticamente para comparção com os resultados numéricos... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: One of the most important features of non-Abelian gauge theories is the non-linearity of the classical field equations. In the context of Yang-Mills theory it is shown that this feature can cause the gauge field to show chaotic behavior. That can happen even when we are considering the field dynamics in the absence of sources, i.e., the vacuum of the Yang-Mills theory. We discuss the connection between chaotic and ergodic behaviors. Then we introduce Berdichevsky's formulation of Classical Statistical Mechanics for Hamiltonian dynamical systems that are both ergodic and low-dimensional. Berdichevsky's theory of Statistical Mechanics is used to study the simplest situation in a non-Abelian gauge theory where the field variables are chaotic and the corresponding phase space has the necessary geometric property. We show that, for the purposes of this study, a pair of complex scalar fields must be introduced in the problem. More precisely, we analyse the so-called non-Abelian Higgs model; the Lagrangian of the theory we are considering has a SU(2) symmetry. The transition from a non-equilibrium dynamical description of the Yang-Mills-Higgs system to a thermodynamical description when it reaches equilibrium is numerically investigated. It is shown that after a sufficiently long time the numerical solutions behave in such a manner that the system can be described by quantities like the temperature, determined in accordance with the prescriptions of equilibrium Statistical Mechanics. These are predicted analytically for comparison with the numerical results. It is verified that there is agreement between analytical and numerical predictions so that the thermalization of the Yang-Mills-Higgs system can be explained with the aid of Berdichevsky's Statistical Mechanics. A dynamical approach to the study... (Complete abstract click electronic access below) / Mestre

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI Company.

Η παρούσα διατριβή έχει ως θέμα της το κβαντικό χάος σε μπιλλιάρδα. Ειδικότερα, εισάγεται ένας μετασχηματισμός (Μετασχηματισμός Εφελκυσμού), ο οποίος προβάλλει το σύνορο ενός μπιλλιάρδου πάνω στον μοναδιαίο κύκλο. Αυτό εισάγει μια μη-Ευκλείδια μετρική στο επίπεδο και έναν διαφορικό τελεστή, ο οποίος περιέχει όλη την πληροφορία σχετικά με το σχήμα του συνόρου και τις ιδιότητες, ως προς την ολοκληρωσιμότητα ή μη, του μπιλλιάρδου. Κλασικά οι ευθείες γραμμές της ελεύθερης κίνησης αντιστοιχούν σε γεωδαισιακές, και κβαντομηχανικά το ενεργειακό φάσμα είναι αυτό του τελεστή Laplace-Beltrami με Dirichlet συνοριακές συνθήκες στον μοναδιαίο κύκλο. Οι γεωδαισιακές εξισώσεις είναι μη-γραμμικές, ομως στο διάστημα μεταξύ δύο διαδοχικών σκεδάσεων υπάρχουν δύο ολοκληρώματα κίνησης, αυτό της κινητικής ενέργειας και αυτό της στροφορμής, οπότε είναι δυνατή η λύση τους. Οι λύσεις αυτές μπορούν να χρησιμοποιηθούν στο κλασικό πρόβλημα σκέδασης. Κβαντικά παίρνουμε το φάσμα των μπιλλιάρδων: Έλλειψη, στάδιο, Robnik και τετράγωνο, για διάφορες τιμές μιας παραμέτρου διαταραχής. Το φάσμα υπολογίζεται διαταρακτικά για μικρές τιμές της παραμέτρου διαταραχής και με διαγωνοποίηση για πιο μεγάλες τιμές της. Η μέθοδος αυτή μπορεί να εφαρμοστεί σε οποιοδήποτε σχήμα συνόρου μπιλλιάρδου, αρκεί ο μετασχηματισμός να είναι αντιστρέψιμος, και μπορεί να χρησιμοποιηθεί σαν ένας γρήγορος τρόπος προσδιορισμού του φάσματος καθώς και σαν ένα θεωρητικό εργαλείο για την ανάλυση θεμελιακών ιδιοτήτων της ολοκληρωσιμότητας, του χάους και της ενδιάμεσης αυτών περιοχής, μέσω του τελεστή Laplace-Beltrami. Σαν ένδειξη των δυνατοτήτων της μεθόδου παραθέτουμε ένα γραφικό τεστ, όπου για πολύ μικρές αποκλίσεις από τον μοναδιαίο κύκλο ένα ολοκληρώσιμο και δύο εν-δυνάμει χαοτικά μπιλλιάρδα διακρίνονται καθαρά μεταξύ τους από τις κατανομές των διαφορών της διόρθωσης πρώτης τάξης στην ενέργεια. Το τεστ αυτό εμφανίζεται για πρώτη φορά στη βιβλιογραφία και έρχεται να συμπληρώσει την γνωστή κατανομή αποστάσεων εγγυτάτων γειτόνων, η οποία για τόσο μικρές αποκλίσεις από το κυκλικό μπιλλιάρδο δεν καταφέρνει να διαχωρίσει τα ολοκληρώσιμα από τα μη-ολοκληρώσιμα σχήματα. Τέλος εισάγεται η έννοια του ανοικτού μπιλλιάρδου, στο οποίο θεωρείται ότι το σύνορο βρίσκεται στο άπειρο. Τα ανοικτά μπιλλιάρδα αν και είναι ολοκληρώσιμα, περιέχουν εντούτοις την πληροφορία για την ολοκληρωσιμότητα ή μη των αντιστοίχων κλειστών σχημάτων. Για την εξαγωγή της τελευταίας πληροφορίας χρησιμοποιούνται διάφοροι μέθοδοι όπως συναρτήσεις αυτο- και ετερο- συσχέτισης. / For a billiard of a general shape a transformation is introduced (Stretching Transformation) which projects the boundary on the unit circle. This introduces a non-Euclidean metric on the plane, which contains all relevant information of the shape of the boundary. Classically the straight lines of the free motion correspond to geodesics and quantum mechanically the energy spectrum is that of Laplace-Beltrami operator with Dirichlet boundary conditions on the unit circle. The geodesic equations are highly non-linear. Nevertheless for the interval between two consecutive scatterings we have two integrals of motion, the kinetic energy and the angular momentum. This fact helps to solve explicitly the geodesic equations. These solutions can be used to derive interesting properties for the classical scattering. Quantum mechanically the spectrum of the above billiards is obtained for certain parameter values both perturbatively for small values of the parameter and also using a diagonalisation procedure. This method is applicable to any particular form of a billiard for which the transformation is invertible and can be used on one hand as a quick method of approximate spectral determination and as a theoretical tool to analyze specific properties of integrability and chaos through the associated connection form and the Laplace-Beltrami operator. As aν indication of the potentiality of this method we present a graphical test where for very small deviations from the circular billiard an integrable and two non-integrable billiards can be distinguished by the distribution of the differences of the first order corrections while this distinction is not evident by the usual test for the nearest neighbor level spacing. Furthermore the open billiard concept is being introduced. An open billiard is one whose boundary is assumed to be at infinity, thus being classified as an integrable billiard, which contains nevertheless the information about potential non-integrability within. Various methods for the extraction of this hidden information are being investigated.

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Quantum chaos has emerged in the half of the last century with the notorious problem of scattering of heavy nuclei. Since then, theoreticians have developed powerful techniques to approach disordered quantum systems. In the late 70's, Casati and Chirikov initiated a new field of research by studying the quantum counterpart of classical problems that are known to exhibit chaos. Among the several quantum-classical chaotic systems studied, the kicked rotor stimulated a lot of enthusiasm in the scientific community due to its equivalence to the Anderson tight binding model. This equivalence allows one to map the random Anderson model into a set of fully deterministic equations, making the theoretical analysis of Anderson localization considerably simpler. In the one-dimensional linear regime, it is known that Anderson localization always prevents the diffusion of the momentum. On the other hand, for higher dimensions it was demonstrated that for certain conditions of the disorder parameter, Anderson localized modes can be inhibited, allowing then a phase transition from localized (insulating) to delocalized (metallic) states. In this thesis we will numerically and theoretically investigate the properties of a multidimensional quantum kicked rotor in a nonlinear medium. The presence of nonlinearity is particularly interesting as it raises the possibility of having soliton waves as eigenfunctions of the systems. We keep the generality of our approach by using an adjustable diffusive nonlinearity, which can describe several physical phenomena. By means of Variational Calculus we develop a chaotic map which fully describes the soliton dynamics. The analysis of such a map shows a rich physical scenario that evidences the wave-particle behavior of a soliton. Through the nonlinearity, we trace a correspondence between quantum and classical mechanics, which has no equivalent in linearized systems. Matter waves experiments provide an ideal environment for studying Anderson localization, as the interactions in these systems can be easily controlled by Feshbach resonance techniques. In the end of this thesis, we propose an experimental realization of the kicked rotor in a dipolar Bose Einstein Condensate.

The Floquet operator, defined as the time-evolution operator over one period, plays a central role in the work presented in this thesis on periodically perturbed quantum systems. Knowledge of the spectral nature of the Floquet operator gives us information on the dynamics of such systems. The work presented here on the spectrum of the Floquet operator gives further insight into the nature of chaos in quantum mechanics. After discussing the links between the spectrum, dynamics and chaos and pointing out an ambiguity in the physics literature, I present a number of new mathematical results on the existence of different types of spectra of the Floquet operator. I characterise the conditions for which the spectrum remains pure point and then, on relaxing these conditions, show the emergence of a continuous spectral component. The nature of the continuous spectrum is further analysed, and shown to be singularly continuous. Thus, the dynamics of these systems are a candidate for classification as chaotic. A conjecture on the emergence of a continuous spectral component is linked to a long standing number-theoretic conjecture on the estimation of finite exponential sums.