We are interested in the onset of instability of the axisymmetric flow between two concentric spherical shells that differentially rotate about a common axis in the narrow-gap limit. The expected mode of instability takes the form of roughly square axisymmetric Taylor vortices which arise in the vicinity of the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radii. At the heart of the difficulties faced is the presence of phase mixing in the system, characterised by a non-zero frequency gradient at the equator and the tendency for vortices located off the equator to oscillate. This mechanism serves to enhance viscous dissipation in the fluid with the effect that the amplitude of any initial disturbance generated at onset is ultimately driven to zero. In this thesis we study a complex Ginzburg-Landau equation derived from the weakly nonlinear analysis of Harris, Bassom and Soward [D. Harris, A. P. Bassom, A. M. Soward, Global bifurcation to travelling waves with application to narrow gap spherical Couette flow, Physica D 177 (2003) p. 122-174] (referred to as HBS) to govern the amplitude modulation of Taylor vortex disturbances in the vicinity of the equator. This equation was developed in a regime that requires the angular velocities of the bounding spheres to be very close. When the spherical shells do not co-rotate, it has the remarkable property that the linearised form of the equation has no non-trivial neutral modes. Furthermore no steady solutions to the nonlinear equation have been found. Despite these challenges Bassom and Soward [A. P. Bassom, A. M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid Mech. 499 (2004) p. 277-314] (referred to as BS) identified solutions to the equation in the form of pulse-trains. These pulse-trains consist of oscillatory finite amplitude solutions expressed in terms of a single complex amplitude localised as a pulse about the origin. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and the whole pulse-train is modulated under an envelope and drifts away from the equator at a relatively slow speed. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; as the absence of steady solutions suggests, self-interaction is inadequate. Though we report new solutions to the HBS co-rotation model the primary focus in this work is the physically more interesting case when the shell velocities are far from close. More specifically we concentrate on the investigation of BS-style pulse-train solutions and, in the first part of this thesis, develop a generic framework for the identification and classification of pulse-train solutions. Motivated by relaxation oscillations identified by Cole [S. J. Cole, Nonlinear rapidly rotating spherical convection, Ph.D. thesis, University of Exeter (2004)] whilst studying the related problem of thermal convection in a rapidly rotating self-gravitating sphere, we extend the HBS equation in the second part of this work. A model system is developed which captures many of the essential features exhibited by Cole's, much more complicated, system of equations. We successfully reproduce relaxation oscillations in this extended HBS model and document the solution as it undergoes a series of interesting bifurcations.

No description available.

<p>Chapter 1 to 4 give a short introduction to superconductivity, microscopic theory, phase transitions, and Monte-Carlo simulations. Chapter 2 is about Cooper pairing in different settings, but I also give a short introduction to the Hofstadter problem of lattice fermions on a square lattice in a perpendicular magnetic field. The purpose is to clarify some points in Paper-I. Chapter 3 is about phase transitions, and introduces the important concepts of spontaneous symmetry breaking, scaling, and renormalization. In the last section I stress some of the main differences between first order and second order phase transitions. Chapter 4 starts with a short elementary introduction to Monte-Carlo simulations and proceeds with the important, but somewhat more advanced topic of reweighting.</p><p>Chapter 5 to 7 are more closely related to the specific projects I have worked on, and are meant to illuminate and clarify some aspects in Paper-II and Paper-III. Chapter 5 introduce the Ginzburg-Landau model in various parametrizations, present some perturbative (mean-field) results, and introduce the concept of topological defects (vortices) and duality.</p><p>Chapter 6 is closely related to Paper-II and introduce the concept of fractal dimension and the relation between the vortex excitations of the original theory and the dual field theory. Chapter 7 is closely related to Paper-III where we studied the order of the metal to superconductor phase transition. To do this we had to do infinite volume and continuum limit extrapolations. We also had to consider ultraviolet renormalization since the Ginzburg-Landau theory is a continuum field theory with no inherent short scale cut-off. To reduce auto-correlation times we added several improvements to the standard Metropolis algorithm in the Monte-Carlo simulations, the most important being an overrelaxation algorithm for the scalar field and a global update of the scalar amplitude.</p>

Chapter 1 to 4 give a short introduction to superconductivity, microscopic theory, phase transitions, and Monte-Carlo simulations. Chapter 2 is about Cooper pairing in different settings, but I also give a short introduction to the Hofstadter problem of lattice fermions on a square lattice in a perpendicular magnetic field. The purpose is to clarify some points in Paper-I. Chapter 3 is about phase transitions, and introduces the important concepts of spontaneous symmetry breaking, scaling, and renormalization. In the last section I stress some of the main differences between first order and second order phase transitions. Chapter 4 starts with a short elementary introduction to Monte-Carlo simulations and proceeds with the important, but somewhat more advanced topic of reweighting. Chapter 5 to 7 are more closely related to the specific projects I have worked on, and are meant to illuminate and clarify some aspects in Paper-II and Paper-III. Chapter 5 introduce the Ginzburg-Landau model in various parametrizations, present some perturbative (mean-field) results, and introduce the concept of topological defects (vortices) and duality. Chapter 6 is closely related to Paper-II and introduce the concept of fractal dimension and the relation between the vortex excitations of the original theory and the dual field theory. Chapter 7 is closely related to Paper-III where we studied the order of the metal to superconductor phase transition. To do this we had to do infinite volume and continuum limit extrapolations. We also had to consider ultraviolet renormalization since the Ginzburg-Landau theory is a continuum field theory with no inherent short scale cut-off. To reduce auto-correlation times we added several improvements to the standard Metropolis algorithm in the Monte-Carlo simulations, the most important being an overrelaxation algorithm for the scalar field and a global update of the scalar amplitude.

Molts sistemes físics tenen la propietat que la seva dinàmica ve definida per algun tipus de difussió espaial en competició amb un fenòmen de reacció, com per exemple en el cas de dos components químics que reaccionen al mateix temps que es difon l'un en el si de l'altre. La presència d'aquests dos fenòmens, la difusió i la reacció, sovint dóna lloc a patrons no homogenis de gran riquesa. Els models matemàtics que descriuen aquest tipus de comportament són normalment equacions en derivades parcials les solucions de les quals representen aquests patrons. En aquesta tesi s'analitza l'equació de Ginzburg-Landau complexa general, que és una equació en derivades parcials de reacció-difusió que s'utilitza sovint com a model matemàtic per a descriure sistemes oscil·latoris en dominis extensos. En particular estudiem els patrons que sorgeixen en el pla quan s'imposa que el grau de Brouwer de la solució no sigui nul. Aquests patrons estan formats per ones de rotació en forma d'espirals, és a dir, les corbes de nivell de la solució formen espirals que emanen dels punts on la funció s'anul·la. Quan la solució s'anul·la només en un punt i per tant només hi ha una espiral, tota la dependència temporal apareix en el terme de freqüència. Així doncs, la funció solució es pot expressar com a funció del radi polar i en termes del seu grau topològic i la freqüència de l'ona. Per tant, aquestes solucions es poden expressar en termes d'un sistema d'equacions diferencials ordinàries. Aquestes solucions només existeixen per una certa freqüència que depèn unívocament dels paràmetres de l'equació i, com a conseqüència i degut a la relació de dispersió entre el nombre d'ones i la freqüència, el nombre d'ones a l'infinit, l'anomenat nombre d'ones asimptòtic, ve també determinat unívocament pels paràmetres. Quan les solucions tenen més d'un zero aïllat la condició sobre el grau de la funció fa que de cada zero sorgeixi una espiral diferent i aquestes es mouen en el pla mantenint la seva estructura local. En aquest treball s'usen tècniques d'anàlisi asimptòtica per trobar equacions del moviment per als centres de les espirals i es troba que aquesta evolució temporal és lenta. En concret, per la distàncies relatives grans entre els centres de les espirals, l'escala de temps per a la seva dinàmica ve donada pel logaritme de l'invers d'aquesta distància. Es demostra que aquestes equacions del moviment són diferents en funció de la relació entre els paràmetres de l'equació de Ginzburg-Landau complexa i la separació entre els centres de les espirals, i que la forma com es passa d'unes equacions a les altres és molt singular. També es demostra que el nombre d'ones asimptòtic per al cas de sistemes amb diverses espirals també està unívocament determinat pels paràmetres però no obstant, el cas de sistemes amb diverses espirals es diferencia del cas d'una única ona en què deixa de ser constant i evoluciona al mateix ritme que la velocitat dels centres de les espirals. / Many physical systems have the property that its dynamics is driven by some kind of spatical diffusion that is in competition with a reaction, like for instance two chemical species that react at the same time that there is a diffusion of each of them into the other. This interplay between reaction and diffusion produce non-homogeneous patterns that can sometimes be very rich. The mathematical models that describe this kind of behaviours are usually nonlinear partial differential equations whose solutions represent these patterns. In this thesis we focus on an especific reaction-diffusion equation that is the so-called general complex Ginzburg-Landau equation that is used as a model for oscillatory systems in extended domains. In particular we are interested in the type of patterns in the plane that arise when the solutions have a non-vanishing Brouwer degree. These patterns have the property that they exhibit rotating waves in the shape of spirals, which means that the contour lines arrange in the shape of spirals that emerge from the points where the solution vanishes. When the solution vanishes only at one point all the time dependence appears as a frequency term so the solutions can be expressed as a function of the polar radius and in terms of the topological degree of the solution and the frequency of the wave. Therefore, these solutions can be expressed in terms of a system of ordinary differential equations. These solutions do only exist with a given frequency, and as a consequence and due to the existence of a dispresion relation, the wavenumber far from the origin, the so-called asymptotic wavenumber, is also unique. When the solutions have more than one isolated zero, the condition on the degree of the function has the effect of producing several spirals that emerge from the different zeros of the solution. These spirals evolve in time keeping their structure but moving around on the plane. In this work we use asymptotic analysis techniques to derive laws of motion for the centres of the spirals and we show that the time evolution of these patterns is slow and, for large relative separations of the centres of the spirals, the time scale for the their dynamics is logarithmic in the inverse of this distance. These laws of motion are different depending on the relation between the parameters of the complex Ginzburg-Landau equation and the relative separation of the spirals. We show that the way these laws change as the spirals separate or approach is highly singular. We also show that the asymptotic wavenumber in the case of multiple spirals is as well unique and that it evolves in time at the same rate as the velocity of the centres.

Resumo: O interesse em conhecer o comportamento supercondutor tem sido cada vez maior nas últimas décadas. Na busca de melhores características supercondutoras, descobriu-se que amostras volumétricas apresentam características muito diferentes de amostras mesoscópicas (amostras com dimensões próximas dos comprimentos de penetração de London e coerência). Como exemplo, podemos citar a não formação de rede de Abrikosov, como consequência do efeito de confinamento (efeito associado às dimensões reduzidas da amostra) e também uma mudança considerável nos valores dos campos críticos. Neste trabalho foram resolvidas as equações de Ginzburg-Landau dependentes do tempo (TDGL), para fazer uma análise detalhada da dinâmica de vórtices em filmes finos mesoscópicos. Para revolvê-las, utilizamos o método das variáveis de ligação com invariância de calibre, adaptado para o algoritmo de diferenças finitas, utilizado para obter a densidade dos pares de Cooper e também curvas de magnetização. O estudo dessa dinâmica de vórtices, foi feito em três amostras com superfícies geométricas diferentes (côncova, convexa e rugosa). Observamos que na comparação entre as duas primeiras, há uma diferença considerável nos valores dos campos críticos, bem como no comportamento da magnetização comparado com um filme plano. Já para a amostra de superfície rugosa, observamos que existe uma competição entre o efeito de confinamento e a rugosidade em relação à configuração dos vórtices. Apresentamos também, uma tabela que mostra resumidamente os estados estacionários dos vórtices nas três amostras. / Abstract: The interest to investigate the investigate the behavior of a superconductor has grown in the last few decades. Having in mind to search for better superconducting characteristics, it has been found that bulk samples present characteristics much more different than mesoscopic samples (samples with dimensions of the same order of the same order of the London penetration length and the coherence length). As an example, we can mention the non-formation of an Abrikosov vortex lattice as a consequence of the confinement effect (effect associated with the reduced dimensions of the sample) and also considerable change in the critical field values. In the present work we solved the time dependent Ginzburg-Landau equation (TDGL), in order to make a detailed analysis of the vortex dynamics in mesoscopic thin films. To solve these equations, we have used the link variables method which is gauge invariant. From this, we obtain the Cooper pair density and the magnetization curves. The vortex dynamics was investigated for three different surfaces of the film (concave, convex, and irregular). We have observed that, with respect to the parabolic geometries, there is a considerable difference for the critical fields, as well as for the behavior of the magnetization compared to a flat film. On the other hand, for a sample with an irregular surface, we have seen that there is a competition between the confinement effect and rugosity with respect to vortex configurations. We also present a table which summarizes the vortex stationary states for the three topologies mentioned above. / Orientador: Paulo Noronha Lisboa Filho / Coorientador: Edson Sardella / Banca: Wilson Aires Ortiz / Banca: Clelio Clemente de Souza Silva / Mestre

Made available in DSpace on 2014-08-27T14:36:44Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-03-27Bitstream added on 2014-08-27T15:57:04Z : No. of bitstreams: 1 000778489.pdf: 2941624 bytes, checksum: caafca05fcfe2ec15edf485aae9368e1 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesta dissertação, propriedades de equilíbrio e não equilíbrio termodinâmico do setor de quarks leves u e d da Cromodinâmica Quântica (QCD) são estudadas empregando o modelo Polyakov– Nambu–Jona-Lasinio(PNJL). O modelo PNJL permite considerar simultaneamente as transições de fase quiral e de desconfinamento à temperatura finita. O grande potencial termodinâmico do modelo foi calculado na aproximação de campo médio. As equações de gap para os parâmetros de ordem que caracterizam essas transições de fase, o condensado de quarks e o loop de Polyakov, foram resolvidas numericamente para diferentes temperaturas e a natureza das transições de fase associadas foi determinada. A seguir,foram obtidas as equações de Ginzburg-Landau-Langevin (GLL) que descrevem a dinâmica temporal dos parâmetros de ordem. As escalas de tempo envolvidas na termalização do condensado de quark e do loop de Polyakov após o sistema ser submetido a um quench de temperatura foram investigadas como função dos parâmetros de Onsager para a QCD. A relevância dos resultados obtidos na presente dissertação para experimentos de colisões de íons pesados a altas energias é dicutida / Thermodynamic equilibrium and non-equilibrium properties of the light u and d quarks sector of Quantum Chromodynamics (QCD) are studied with the Polyakov–Nambu–Jona-Lasinio (PNJL) model. The PNJL model allows to take into account simultaneously the chiral and deconfinement transitions at finite temperatures. The gran potential of the model is obtained in the mean field approximation. The gap equations for the order parameters that characterise these transitions, the quark condensate and the Polyakov loop, are solved numerically for different temperatures and the nature of the associated phase transitions is determined. Next, the Ginzburg-Landau-Langevin (GLL) equations that describe the temporal dynamics of the order parameters are obtained. The time scales involved in the thermalization of the quark condensate and Polyakov loop after a temperature quench are investigated as functions of the QCD Onsager parameters available in the literature. The relevance of the results obtained in the present dissertation for experiments of heavy ions collisions at high energies are discussed

Orientador: Gastão Inácio Krein / Banca: Marcus Emmanuel Benghi Pinto / Banca: Ricardo D'Elia Matheus / Resumo: Nesta dissertação, propriedades de equilíbrio e não equilíbrio termodinâmico do setor de quarks leves u e d da Cromodinâmica Quântica (QCD) são estudadas empregando o modelo Polyakov- Nambu-Jona-Lasinio(PNJL). O modelo PNJL permite considerar simultaneamente as transições de fase quiral e de desconfinamento à temperatura finita. O grande potencial termodinâmico do modelo foi calculado na aproximação de campo médio. As equações de gap para os parâmetros de ordem que caracterizam essas transições de fase, o condensado de quarks e o loop de Polyakov, foram resolvidas numericamente para diferentes temperaturas e a natureza das transições de fase associadas foi determinada. A seguir,foram obtidas as equações de Ginzburg-Landau-Langevin (GLL) que descrevem a dinâmica temporal dos parâmetros de ordem. As escalas de tempo envolvidas na termalização do condensado de quark e do loop de Polyakov após o sistema ser submetido a um quench de temperatura foram investigadas como função dos parâmetros de Onsager para a QCD. A relevância dos resultados obtidos na presente dissertação para experimentos de colisões de íons pesados a altas energias é dicutida / Abstract: Thermodynamic equilibrium and non-equilibrium properties of the light u and d quarks sector of Quantum Chromodynamics (QCD) are studied with the Polyakov-Nambu-Jona-Lasinio (PNJL) model. The PNJL model allows to take into account simultaneously the chiral and deconfinement transitions at finite temperatures. The gran potential of the model is obtained in the mean field approximation. The gap equations for the order parameters that characterise these transitions, the quark condensate and the Polyakov loop, are solved numerically for different temperatures and the nature of the associated phase transitions is determined. Next, the Ginzburg-Landau-Langevin (GLL) equations that describe the temporal dynamics of the order parameters are obtained. The time scales involved in the thermalization of the quark condensate and Polyakov loop after a temperature quench are investigated as functions of the QCD Onsager parameters available in the literature. The relevance of the results obtained in the present dissertation for experiments of heavy ions collisions at high energies are discussed / Mestre

Made available in DSpace on 2014-06-11T19:30:19Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-04-28Bitstream added on 2014-06-13T19:39:47Z : No. of bitstreams: 1 pascolati_mcv_me_bauru.pdf: 4047066 bytes, checksum: cf327b8f4cd0447dc8dd5e7c4d6604a5 (MD5) / O interesse em conhecer o comportamento supercondutor tem sido cada vez maior nas últimas décadas. Na busca de melhores características supercondutoras, descobriu-se que amostras volumétricas apresentam características muito diferentes de amostras mesoscópicas (amostras com dimensões próximas dos comprimentos de penetração de London e coerência). Como exemplo, podemos citar a não formação de rede de Abrikosov, como consequência do efeito de confinamento (efeito associado às dimensões reduzidas da amostra) e também uma mudança considerável nos valores dos campos críticos. Neste trabalho foram resolvidas as equações de Ginzburg-Landau dependentes do tempo (TDGL), para fazer uma análise detalhada da dinâmica de vórtices em filmes finos mesoscópicos. Para revolvê-las, utilizamos o método das variáveis de ligação com invariância de calibre, adaptado para o algoritmo de diferenças finitas, utilizado para obter a densidade dos pares de Cooper e também curvas de magnetização. O estudo dessa dinâmica de vórtices, foi feito em três amostras com superfícies geométricas diferentes (côncova, convexa e rugosa). Observamos que na comparação entre as duas primeiras, há uma diferença considerável nos valores dos campos críticos, bem como no comportamento da magnetização comparado com um filme plano. Já para a amostra de superfície rugosa, observamos que existe uma competição entre o efeito de confinamento e a rugosidade em relação à configuração dos vórtices. Apresentamos também, uma tabela que mostra resumidamente os estados estacionários dos vórtices nas três amostras. / The interest to investigate the investigate the behavior of a superconductor has grown in the last few decades. Having in mind to search for better superconducting characteristics, it has been found that bulk samples present characteristics much more different than mesoscopic samples (samples with dimensions of the same order of the same order of the London penetration length and the coherence length). As an example, we can mention the non-formation of an Abrikosov vortex lattice as a consequence of the confinement effect (effect associated with the reduced dimensions of the sample) and also considerable change in the critical field values. In the present work we solved the time dependent Ginzburg-Landau equation (TDGL), in order to make a detailed analysis of the vortex dynamics in mesoscopic thin films. To solve these equations, we have used the link variables method which is gauge invariant. From this, we obtain the Cooper pair density and the magnetization curves. The vortex dynamics was investigated for three different surfaces of the film (concave, convex, and irregular). We have observed that, with respect to the parabolic geometries, there is a considerable difference for the critical fields, as well as for the behavior of the magnetization compared to a flat film. On the other hand, for a sample with an irregular surface, we have seen that there is a competition between the confinement effect and rugosity with respect to vortex configurations. We also present a table which summarizes the vortex stationary states for the three topologies mentioned above.

No description available.