Dans le cadre d'une approche Higher Tamm Dancoff Approximation notée HTDA, nous avons étudié les corrélations vibrationnelles de type quadrupole, avec et sans appariement. Le champ moyen a été déterminé dans le cadre d'une approche microscopique utilisant l'interaction effective de Skyrme. Une interaction résiduelle schématique de type delta plus quadrupole-quadrupole, tenant compte en particulier de l'appariement neutron-proton T=0 et T=1 a été utilisé. Les résultats obtenus pour la résonance géante quadrupolaire isoscalaire du noyau Ca40 ont été comparés aux données expérimentales et à d'autres résultats théoriques. Nous avons également étudié un gaz de fermions piégés dans un potentiel d'oscillateur harmonique à 2D, et à température nulle, en présence d'un champ magnétique uniforme. Les expressions exactes des quelques grandeurs thermodynamiques ont été dérivées à partir de la matrice densité de Bloch.

The Sun, by the virtue of its proximity to Earth, serves as an excellent astrophysical laboratory for testing our theoretical ideas. The Sun displays a plethora of visually awe-inspiring phenomena including flares, prominences, sunspots, corona, CMEs and uncountable others. It is now known that it is the magnetic field of the Sun which governs all these and also the geomagnetic storms at the Earth, which owes its presence to the interaction between the geomagnetic field and the all-pervading Solar magnetic field in the interplanetary medium. Since the solar magnetic field affects the interplanetary space around the Earth in a profound manner, it is absolutely essential that we develop a comprehensive understanding of the generation and manifestation of magnetic fields of the Sun. This thesis aims at developing a state-of-the-art dynamo code SURYA1taking into account important results from helioseismology and magnetohydrodynamics. This dynamo code is then used to study various phenomenon associated with solar activity including evolution of solar parity, response to stochastic fluctuations, helicity of active regions and prediction of future solar cycles. Within last few years dynamo theorists seem to have reached a consensus on the basic characteristics of a solar dynamo model. The solar dynamo is now believed to be comprised of three basic processes: (i)The toroidal field is produced by stretching of poloidal field lines primarily inside the tachocline – the region of strong radial shear at the bottom of the convection zone. (ii) The toroidal field so formed rises to the surface due to magnetic buoyancy to form active regions. (iii) Poloidal field is generated at the surface due to decay of tilted active regions – an idea attributed to Babcock (1961) & Leighton (1969). The meridional circulation then carries the poloidal field produced near the surface to the tachocline. The profile of the solar differential rotation has now been mapped by helioseismology and so has been the poleward branch of meridional circulation near the surface. The model I describe in this thesis is a two-dimensional kinematic solar dynamo model in a full sphere. Our dynamo model Surya was developed over the years in stages by Prof. Arnab Rai Choudhuri, Dr. Mausumi Dikpati, Dr. Dibyendu Nandy and myself. We provide all the technical details of our model in Chap. 2 of this thesis. In this model we assume the equatorward branch of the meridional circulation (which hasn’t been observed yet), to penetrate slightly below the tachocline (Nandy & Choudhuri 2002, Science, 296, 1671). Such a meridional circulation plays an important role in suppressing the magnetic flux eruptions at high latitudes. The only non-linearity included in the model is the prescription of magnetic buoyancy. Our model is shown to reproduce various aspects of observational data, including the phase relation between sunspots and the weak, efficient. An important characteristic of our code is that it displays solar-like dipolar parity (anti-symmetric toroidal fields across equator) when certain reasonable conditions are satisfied, the most important condition being the requirement that the poloidal field should diffuse efficiently to get coupled across the equator. When the magnetic coupling between the hemispheres is enhanced by either increasing the diffusion or introducing an α ff distributed throughout the convection zone, we find that the solutions in the two hemispheres evolve together with a single period even when we make the meridional circulation or the α effect different in the two hemispheres. The effect of diffusive coupling in our model is investigated in Chap. 3. After having explored the regular behaviour of the solar cycle using the dynamo code we proceed to study the irregularities of the Solar cycle.We introduce stochastic fluctuations in the poloidal source term at the solar surface keeping the meridional circulation steady for all the numerical experiments. The dynamo displays oscillatory behaviour with variable cycle amplitudes in presence of fluctuations with amplitudes as large as 200%. We also find a statistically significant correlation between the strength of polar fields at the endofone cycle and the sunspot number of the next cycle. In contrast to this there exist a very poor correlation between the sunspot number of a cycle and the polar field formed at its end. This suggests that during the declining phase of the sunspot cycle poloidal field generation from decaying spots takes place via the Babcock-Leighton mechanism which involves randomness and destroys the correlation between sunspot number of a cycle and the polar at its end. In addition to this we also see that the time series of asymmetries in the sunspot activity follows the time series of asymmetries in the polar field strength with a lag of 5 years. We also compare our finding with available observational data. Although systematic measurements of the Sun’s polar magnetic field exist only from mid-1970s, other proxies can be used to infer the polar field at earlier times. The observational data indicate a strong correlation between the polar field at a sunspot minimum and the strength of the next cycle, although the strength of the cycle is not correlated well with the polar field produced at its end. We use these findings about the correlation of polar fields with sunspots to develop an elegant method for predicting future solar cycles. We feed observational data for polar fields during the minima of cycle n into our dynamo model and run the code till the next minima in order to simulate the sunspot number curve for cycle n+1. Our results fit the observed sunspot numbers of cycles 21-23 reasonably well and predict that cycle 24 will be about 30–35% weaker than cycle 23. We fit that the magnetic diffusivity in the model plays an important role in determining the magnetic memory of the Solar dynamo. For low diffusivity, the amplitude of a sunspot cycle appears to be a complex function of the history of the polar field of earlier cycles. Only if the magnetic diffusivity within the convection zone is assumed to be high (of order 1012cms−1), we are able to explain the correlation between the polar fiat a minimum and the next cycle. We give several independent arguments that the diffusivity must be of this order. In a dynamo model with diffusivity like this, the poloidal field generated at the mid-latitudes is advected toward the poles by the meridional circulation and simultaneously diffuses towards the tachocline, where the toroidal field for the next cycle is produced. The above ideas are put forward in Chap. 6. We next come to an important product of the dynamo process namely the magnetic helicity. It has been shown independently by many research groups that the mean value of the normalized current helicity αp= B (Δ×B)/B2in solar active regions is of the order of 10−8m−1, predominantly negative in the northern hemisphere, positive in the southern hemisphere. Choudhuri (2003, Sol. Phys., 215, 31)developed a model for production of the helicity of the required sign in a Babcock-Leighton Dynamo by wrapping of poloidal field lines around a fluxtube rising through the convection zone. In Chap. 7 we calculate helicities of solar active regions based on this idea. Rough estimates based on this idea compare favourably with the observed magnitude of helicity. We use our solar dynamo model to study how helicity varies with latitude and time. At the time of solar maximum, our theoretical model gives negative helicity in the northern hemisphere and positive helicity in the south, in accordance with observed hemispheric trends. However, we fit that during a short interval at the beginning of a cycle, helicities tend to be opposite of the preferred hemispheric trends. After calculating the sign and magnitude of helicity of the sunspots we worry about the distribution of helicity inside a sunspot. In Chap. 8 we model the penetration of a wrapped up background poloidal field into a toroidal magnetic flux tube rising through the solar convective zone. The rise of the straight, cylindrical flux tube is followed by numerically solving the induction equation in a comoving Lagrangian frame, while an external poloidal magnetic field is assumed to be radially advected onto the tube with a speed corresponding to the rise velocity. One prediction of our model is the existence of a ring of reverse current helicity on the periphery of active regions. On the other hand, the amplitude of the resulting twist depends sensitively on the assumed structure (ffvs. concentrated/intermittent) of the active region magnetic field right before its emergence, and on the assumed vertical profile of the poloidal field. Nevertheless, in the model with the most plausible choice of assumptions a mean twist comparable to the observational results. Our results indicate that the contribution of this mechanism to the twist can be quite find under favourable circumstances it can potentially account for most of the current helicity observed in active regions.

No description available.

De nos jours, il est bien accepté que le cycle magnétique de 11 ans du Soleil est l'oeuvre d'une dynamo interne présente dans la zone convective. Bien qu'avec la puissance de calculs des ordinateurs actuels il soit possible, à l'aide de véritables simulations magnétohydrodynamiques, de résoudre le champ magnétique et la vitessse dans toutes les directions spatiales, il n'en reste pas moins que pour étudier l'évolution temporelle et spatiale de la dynamo solaire à grande échelle, il reste avantageux de travailler avec des modèles plus simples. Ainsi, nous avons utilisé un modèle simplifié de la dynamo solaire, nommé modèle de champ moyen, pour mieux comprendre les mécanismes importants à l'origine et au maintien de la dynamo solaire. L'insertion d'un tenseur-alpha complet dans un modèle dynamo de champ moyen, provenant d'un modèle global-MHD [Ghizaru et al., 2010] de la convection solaire, nous a permis d'approfondir le rôle que peut jouer la force électromotrice dans les cycles magnétiques produits par ce modèle global. De cette façon, nous avons pu reproduire certaines caractéristiques observées dans les cycles magnétiques provenant de la simulation de Ghizaru et al., 2010. Tout d'abord, le champ magnétique produit par le modèle de champ moyen présente deux modes dynamo distincts. Ces modes, de périodes similaires, sont présents et localisés sensiblement aux mêmes rayons et latitudes que ceux produits par le modèle global. Le fait que l'on puisse reproduire ces deux modes dynamo est dû à la complexité spatiale du tenseur-alpha. Par contre, le rapport entre les périodes des deux modes présents dans le modèle de champ moyen diffère significativement de celui trouvé dans le modèle global. Par ailleurs, on perd l'accumulation d'un fort champ magnétique sous la zone convective dans un modèle où la rotation différentielle n'est plus présente. Ceci suggère que la présence de rotation différentielle joue un rôle non négligeable dans l'accumulation du champ magnétique à cet endroit. Par ailleurs, le champ magnétique produit dans un modèle de champ moyen incluant un tenseur-alpha sans pompage turbulent global est très différent de celui produit par le tenseur original. Le pompage turbulent joue donc un rôle fondamental au sein de la distribution spatiale du champ magnétique. Il est important de souligner que les modèles dépourvus d'une rotation différentielle, utilisant le tenseur-alpha original ou n'utilisant pas de pompage turbulent, parviennent tous deux à produire une dynamo oscillatoire. Produire une telle dynamo à l'aide d'un modèle de ce type n'est pas évident, a priori. Finalement, l'intensité ainsi que le type de profil de circulation méridienne utilisés sont des facteurs affectant significativement la distribution spatiale de la dynamo produite. / It is generally agreed upon that the 11-year magnetic cycle of the Sun arises through the action of an internal dynamo operating in the convective zone, and perhaps also immediately beneath it. Although the computing power of current supercomputers is sufficient to allow fairly realistic magnetohydrodynamical simulations of this dynamo process, to study the temporal and spatial evolution of the large-scale solar magnetic field over long timescales, it remains advantageous to work with simpler models. Thus, to better understand the physical mechanisms at the origin and maintenance of the solar dynamo, we used a simplified formulation, known as a mean-field model. By using a complete alpha-tensor extracted from a global MHD model of solar convection [Ghizaru et al., 2010] as input to a kinematic axisymmetric mean-field dynamo model [Charbonneau & MacGregor, 1997], it becomes possible to study the effect of the electromotive force on the magnetic cycles produced by the global model. In this way, we are able to reproduce some of the observed characteristics of the Ghizaru et al., 2010 simulation, in particular magnetic cycles. The axisymmetric magnetic field produced by the mean-field dynamo model exhibits two distincts dynamo modes. These modes, with similar periods, are present and peak at substantially at the same radii and latitudes as the sonlly-averaged magnetic fields extracted from the global model. Thanks to the spatial complexity of the alpha-tensor, we can reproduce these two dynamo modes. In contrast, the ratio of the periods of the two modes present in the mean field model differs significantly from that found in the global model. In addition, the accumulation of strong magnetic fields at the base of the convective zone disappears in a model where differential rotation has been removed. This suggests that differential rotation plays a significant role in the accumulation of magnetic fields in this region. Furthermore, removing the turbulent pumping component of the alpha-tensor produces a very different magnetic field cycle. Therefore, turbulent pumping plays a crucial role in the spatial distribution of the magnetic field. It is important to underline that the models without differential rotation, with or without turbulent pumping, both succeed in producing an oscillatory dynamo using only the turbulent electromotive force. However, the dynamos materializing in these modified models are significantly different from that using the full alpha-tensor. Finally, both the intensity and form of meridional circulation profiles are significant factors affecting the dynamo modes.

In the first part of this thesis, motivated by the development of deep brain stimulation for Parkinson's disease, we consider the problem of reducing the synchrony of a neuronal population via a closed-loop electrical stimulation. This, under the constraints that only the mean membrane voltage of the ensemble is measured and that only one stimulation signal is available (mean-field feedback). The neuronal population is modeled as a network of interconnected Landau-Stuart oscillators controlled by a linear single-input single-output feedback device. Based on the associated phase dynamics, we analyze existence and robustness of phase-locked solutions, modeling the pathological state, and derive necessary conditions for an effective desynchronization via mean-field feedback. Sufficient conditions are then derived for two control objectives: neuronal inhibition and desynchronization. Our analysis suggests that, depending on the strength of feedback gain, a proportional mean-field feedback can either block the collective oscillation (neuronal inhibition) or desynchronize the ensemble.In the second part, we explore two possible ways to analyze related problems on more biologically sound models. In the first, the neuronal population is modeled as the interconnection of nonlinear input-output operators and neuronal synchronization is analyzed within a recently developed input-output approach. In the second, excitability and synchronizability properties of neurons are analyzed via the underlying bifurcations. Based on the theory of normal forms, a novel reduced model is derived to capture the behavior of a large class of neurons remaining unexplained in other existing reduced models.

Ce manuscrit présente les fondements dynamiques des jeux de population avec un nombre variable de joueurs ainsi que leurs concepts de solutions et de stabilités. Nous introduisons d'abord les dynamiques de jeux avec retard et étudions leurs stabilités. Nous les appliquons aux réseaux filaires et aux réseaux sans fils. Ensuite nous nous intéressons aux aspects de mobilité et aux distributions spatiales des joueurs sur le réseau. Cela nous conduit à une nouvelle classe de dynamique de jeux à stratégies vectorielles avec des contraintes de migrations, appelée dynamique de jeux d'évolution avec migration. Nous dérivons de telles dynamiques pour les réseaux hybrides et appliquons aux problèmes de contrôle de puissance dans les réseaux hétérogènes, choix entre plusieurs technologies et migration entre plusieurs classes d'utilisateurs. Ensuite nous nous focalisons aux jeux stochastiques de population avec plusieurs classes de joueurs dans lesquels chaque joueur possède son propre état et fait face un vecteur qui évolue dans le temps. Des applications à la gestion d'énergie dans les réseaux sont présentées. Finalement, nous étudions une classe de jeux à champ moyen. Lorsque la taille de la population devient très grande, les asymptotiques du système conduisent à des dynamiques appelées dynamiques de jeux à champ moyen. Cette classe de dynamiques contient les dynamiques standard basées sur des révisions de stratégies. Nous utilisons ce modèle pour analyser les problèmes accès aléatoires à des ressources dans un environnement où les utilisateurs et les ressources sont spatialement distribuées. Nous établissons un lien entre les jeux à champ moyen et les jeux différentiels de population dans lesquels chaque joueur a son état individuel et optimise son paiement à long terme pendant son temps de séjour dans le système sous contraintes que le profil de population évolue selon une dynamique de jeux à champ moyen / His manuscript presents dynamic foundations of population games with variable number of players and their solutions and stability concepts. We first introduce delayed evolutionary game dynamics and study their stability. Applications to both wired and wireless networks are presented. We then introduce mobility and spatial aspects of players distribution into the network dynamics. This leads to a new class of game dynamics with multicomponent strategies and migration constraints called evolutionary game dynamics with migration. We derived such dynamics for hybrid systems such as power control in heterogenous networks, switching between technologies and migration between different classes of users. After that we focus on stochastic population games with multiple classes of players in which each player has its own state and facing to an evolving vector which represents the population profile. We use this model to analyze resource and energy constrained interactions in wireless networks. Finally, we present a class of mean field games. When taking the asymptotics of finite systems, we derive a new class of game dynamics called mean field game dynamics. This class contains the standard evolutionary game dynamics based on revision of pure actions. We apply this model to analyze spatial random access game and dynamic resource competition game with individual states. We establish a link betweenmean field games and differential population games inwhich each player optimizes its long-term objective during its sojourn time in the system subject to the constraint that the population profile evolves according to some mean field game dynamics

Dans cette thèse on étudie les effets de taille finie au-dessus de la dimension critique supérieure d_c. Les effets de taille finie y ont longtemps été incomplètement compris, en particulier vis-à-vis de leur dépendance en fonction des conditions aux limites. La violation de la relation d’échelle dite d’hyperscaling a été l’un des aspects les plus évidents des difficultés rencontrées. Le désaccord avec le scaling usuel est dû au caractère de variable non pertinente dangereuse du terme de self-interaction dans la théorie en ϕ^4. Celle-ci était considérée comme dangereuse pour la densité d’énergie libre et les fonctions thermodynamiques associées, mais pas dans le secteur des corrélations. Récemment, un schéma nouveau de scaling a été proposé dans lequel la longueur de corrélation joue un rôle central et est également affectée par la variable non pertinente dangereuse. Ce nouveau schéma, appelé QFSS, est basé sur le fait que la longueur de corrélation exhibe au lieu du scaling usuel ξ~L un comportement en puissance de la taille finie ξ~L^ϙ. Ce pseudo-exposant critique ϙ est lié à la dimension critique supérieure et à la variable dangereuse. Au-dessous de d_c, cet exposant prend la valeur ϙ=1, mais au-dessus, il vaut ϙ=d/d_c. Le schéma QFSS est parvenu à réconcilier les exposants de champs moyen et le Finite-Size-Scaling tel que dérivé du Groupe de Renormalisation pour les modèles avec interactions à courte portée au-dessus de d_c en conditions aux limites périodiques. Si ϙ est un exposant universel, la validité de la théorie doit toutefois s’étendre également aux conditions de bords libres. Des tests initiaux dans de telles conditions ont mis en évidence de nouvelles difficultés: alors que le QFSS est valable au point pseudo-critique auquel les grandeurs thermodynamiques telles que la susceptibilité manifestent un pic à taille finie, au point critique on a pensé que c’était le FSS standard qui prévalait avec les exposants de champ moyen et ξ~L. On montre dans ce travail qu’il en va différemment de la situation au point critique et qu’à la place ce sont les exposants gaussiens qui s’appliquent en l’absence de variable non pertinente dangereuse. Pour mettre en évidence ce résultat, nous avons mené des simulations de modèles avec interactions à longue portée, qui peuvent être à volonté étudiés au-dessus de leur dimension critique supérieure. Nous avons aussi développé une étude des modes de Fourier qui permet de fournir des exemples de quantités non affectées par la présence de la variable non pertinente dangereuse / In this project finite-size size scaling above the upper critical dimension〖 d〗_c is investigated. Finite-size scaling there has long been poorly understood, especially its dependency on boundary conditions. The violation of the hyperscaling relation above d_c has also been one of the most visible issues. The breakdown in standard scaling is due to the dangerous irrelevant variables presented in the self-interacting term in the ϕ^4 theory, which were considered dangerous to the free energy density and associated thermodynamic functions, but not to the correlation sector. Recently, a modified finite-size scaling scheme has been proposed, which considers that the correlation length actually plays a pivotal role and is affected by dangerous variables too. This new scheme, named QFSS, considers that the correlation length, instead of having standard scaling behaviour ξ~L , scales as ξ~L^ϙ. This pseudocritical exponent is connected to the critical dimension and dangerous variables. Below d_c this exponent takes the value ϙ=1, but above the upper critical dimension it is ϙ=d/d_c. QFSS succeeded in reconciling the mean-field exponents and FSS derived from the renormalisation-group for the models with short-range interactions above d_c with periodic boundary conditions. If ϙ is an universal exponent, the validity of that theory should also hold for the free boundary conditions. Initial tests for such systems faced new problems. Whereas QFSS is valid at pseudocritical points where quantities such as the magnetic susceptibility experience a peak for finite systems, at critical points the standard FSS seemed to prevail, i.e., mean-field exponents with ξ~L. Here, we show that this last picture at critical point is not correct and instead the exponents that applied there actually arise from the Gaussian fixed-point FSS where the dangerous variables are suppressed. To achieve this aim, we study Ising models with long-range interaction, which can be tuned above〖 d〗_c, with periodic and free boundary conditions. We also include a study of the Fourier modes which can be used as an example of scaling quantities without dangerous variables

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-06-27T20:42:50Z No. of bitstreams: 1 DissLFL.pdf: 818058 bytes, checksum: a45e8f4dbdc692c6f31fde1d45f6574d (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-07-03T17:56:46Z (GMT) No. of bitstreams: 1 DissLFL.pdf: 818058 bytes, checksum: a45e8f4dbdc692c6f31fde1d45f6574d (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-07-03T17:56:52Z (GMT) No. of bitstreams: 1 DissLFL.pdf: 818058 bytes, checksum: a45e8f4dbdc692c6f31fde1d45f6574d (MD5) / Made available in DSpace on 2017-07-03T18:01:59Z (GMT). No. of bitstreams: 1 DissLFL.pdf: 818058 bytes, checksum: a45e8f4dbdc692c6f31fde1d45f6574d (MD5) Previous issue date: 2016-12-20 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / In this thesis, we present a priori estimates for solutions of a mean-field game (MFG) defined over a bounded domain Ω ⊂ ℝd. We propose an application of these results to a model of capital and wealth accumulation. In Chapter 1, an introduction to mean-field games is presented. We also put forward some of the motivation from Economics and discuss previous developments in the theory of differential games. These comments aim at indicating the connection between mean-field games theory, its applications and the realm of Mathematical Analysis. In Chapter 2, we present an optimal control problem. Here, the agents are supposed to be undistinguishable, rational and intelligent. Undistinguishable means that every agent is governed by the same stochastic differential equation. Rational means that all efforts of the agent is to maximize a payoff functional. Intelligent means that they are able to solve an optimal control problem. Once we describe this (stochastic) optimal control problem, we produce a heuristic derivation of the mean-field games system, which is summarized in a Verification Theorem; this gives rise to the Hamilton-Jacobi equation (HJ). After that, we obtain the Fokker-Plank equation (FP). Finally, we present a representation formula for the solutions to the (HJ) equation, together with some regularity results. In Chapter 3, a specific optimal control problem is described and the associated MFG is presented. This MFG is prescribed in a bounded domain Ω ⊂ ℝd, which introduces substantialadditional challenges from the mathematical view point. This is due to estimates for the solutionsat the boundary in Lp. The rest of the chapter puts forward two well known tips of estimates: theso-called Hopf-Lax formula and the First Order Estimate. In Chapter 4, the wealth and capital accumulation mean-field game model is presented. The relevance of studying MFG in a bounded domain then becomes clear. In light of the results obtained in Chapter 3, we close Chapter 4 with the Hopf-Lax formula, and the First Order estimates. Three appendices close this thesis. They gather elementary material on Stochastic Calculus and Functional Analysis. / Nesta dissertação são apresentadas algumas estimativas a priori para soluções de sistemas mean-field games (MFG), definidos em domínios limitados Ω ⊂ ℝd. Tais estimativas são aplicadas em um modelo mean-field específico, que descreve o acúmulo de riqueza e capital. No Capítulo 1, é apresentada uma breve introdução histórica sobre os mean-field games. Nesta introdução, exploramos sua relação com a teoria dos jogos, cujos alicerces foram construídos por economistas e matemáticos ao longo do século XX. O objetivo do capítulo é transmitir. No Capítulo 2, apresentamos um problema de controle ótimo em que cada agente é suposto ser indistinguível, racional e inteligente. Indistinguível no sentido de que cada um é governado pela mesma equação diferencial estocástica. Racional no sentido de que todos os esforços do agente são no sentido de maximizar um funcional de recompensa e, inteligente no sentido de que são capazes de resolver um problema de controle ótimo. Descreve-se este problema de controle ótimo, e apresenta-se a derivação heurística dos mean-field games; obtém-se através de um Teorema de Verificação, a equação de Hamilton-Jacobi (HJ) associada, e em seguida, obtémse a equação de Fokker-Planck. De posse destas equações, apresentamos alguns resultados preliminares, como uma fórmula de representação para soluções da equação de HJ e alguns resultados de regularidade. No Capítulo 3, descreve-se um problema específico de controle ótimo e apresenta-se a respectiva derivação heurística culminando na descrição de um MFG com condições não periódicas na fronteira; esta abordagem é original na literatura de MFG. O restante do capítulo é dedicado à exposição de dois tipos bem conhecidos de estimativas: a fórmula de Hopf-Lax e estimativa de Primeira Ordem. Uma observação relevante, é a de que o trabalho em obter-se estimativas a priori é aumentado substancialmente neste caso, devido ao fato de lidarmos com estimativas para os termos de fronteira com normas em Lp. ao leitor, as origens da Teoria Econômica contemporânea, que surgem à partir da utilização da Matemática na formulação e resolução de problemas econômicos. Tal abordagem é motivada principalmente pelo rigor e clareza da Matemática em tais circunstâncias. No Capítulo 4, apresenta-se o modelo de jogo do tipo mean-field de acúmulo de capital e riqueza, o que deixa claro a relevância do estudo dos MFG em um domínio limitado. À luz dos resultados obtidos no Capítulo 3, encerramos o Capítulo 4 com as estimativas do tipo Hopf-Lax e de Primeira Ordem. Três apêndices encerram o texto desta dissertação de mestrado; estes reúnem material elementar sobre Cálculo Estocástico e Análise Funcional.

A teoria de campo médio para o modelo de Ashkin-Teller com interações ferromagnéticas de longo alcance na presença de campos magnéticos aleatórios foi desenvolvida. Isso foi conseguido através do uso do truque de réplicas para a obtenção da energia livre e do estudo analítico das equações integrais acopladas dos parâmetros de ordem, da estabilidade de suas soluções e das suas expansões para T &#8804 Tc. Inicialmente, foram determinadas as expressões gerais das funções termodinâmicas do modelo no caso em que existiam três campos magnéticos aleatórios com distribuições gaussianas. Em seguida, foi examinado o caso particular do modelo com um só campo magnético aleatório na direção de Z = &#8249 &#948 S &#8250. A estratégia adotada se mostrou poderosa pois possibilitou a caracterização detalhada do diagrama de fases com várias superfícies de coexistência e das linhas de pontos críticos. As equações integrais das funções termodinâmicas desse caso particular foram discutidas e resolvidas numericamente para valores especiais das constantes de interação e da variância. Para o caso particular do modelo na presença de campos magnéticos aleatórios nas direções &#8249 S &#8250 e &#8249 &#948 &#8250, foram determinadas e discutidas as expressões das funções termodinâmicas. Foram também obtidas as equações das superfícies de instabilidade da solução paramagnética. Foi provado que a transição entre as fases paramagnética e de Baxter é sempre de primeira ordem. Outro resultado original da tese foi a verificação da existência da simetria de dilatação e contração do modelo de Potts na presença de campos magnéticos aleatórios. Essa simetria permite que o estudo da energia livre no intervalo q&#8712 (1,2) forneça o comportamento termodinâmico do sistema para todo q>2. / The meanfield theory of the long range Ashkin-Teller model in random fields was developed. This was obtained by using the replica trick and the study of the coupled integral equations for the order parameters, the stability of their solutions, and their expansions for T &#8804 Tc. Inicially, the expressions of the thermodynamic functions for the model in three random fields with Gaussian distributiuons were determined. After this, it was examined the particular case of the model with only one random field in the Z = &#8249 &#948 S &#8250 direction. The strategy revealed itself powerful by the detailed characterization of the phase diagram with several coexistence surfaces and lines of critical points. The integral equations of the thermodynamic functions for this particular case were discussed and numerically solved for special values of the interaction constants and field distribution variance. For the particular case of the model with random fields in the &#8249 S &#8250 and &#8249 &#948 &#8250, directions, the expressions were also determined and discussed. The equations of the instability surfaces for the paramagnetic solution were obtained, and it was proved that the para-Baxter transition line is always of first order. Another original result of the thesis was the verification of the the existence of the dilatation and contration symmetry in the Potts model with random fields. This symmetry permits that the study of the free energy in the q&#8712(1,2) interval supplies the thermodynamics behavior of the system for q>2.

We have modeled some magnetic systems, which consists of a number of sublattices, in the mean field approximation. This is possible in crystalline systems formed by two or more magnetic ions coupled by specific interactions such as the crystal field, exchange, among others. The main idea is to solve the microscopic Hamiltonian that models a given magnetic system in order to obtain their magnetic equation of state: M (H, T). For this, we use the appropriate arrangements to different magnetic (ferro-, ferri- and antiferro- magnetic) scheme sublattices. From the solutions of the Hamiltonian (eigenvalues and eigenvectors), physical quantities of interest were determined. In principle we consider systems with localized magnetism due to 3d and 4f electrons with the participation of non-magnetic ligands including 3d-4f systems with the presence of crystal field. In this dissertation we use the model of two-and three sublattices in order to obtain the equation of state for the following systems: RKKY exchange in RNi2B2C, superexchange in (Y3-zRz)(T1xFe1-x)(T2yFe3-y)O12, LixFe3- xO4 and (NixMn1-x)1.5[Cr(CN)6]. In these formulas, R represents a rare earth ion, T1 and T2 represent non-magnetic ions. Some representative cases are presented to illustrate the different equations of state and behavior of sublattices, metamagnetism, temperature compensation, etc. The extension to other similar systems can be direct or need to incorporate additional phenomenological parameters. / Consideramos neste estudo sistemas magnéticos modeláveis mediante sub- redes na aproximação do campo médio. Isto é possível em sistemas cristalinos formados por dois ou mais íons magnéticos acoplados por interações específicas como as do campo cristalino, troca, entre outros. A ideia central é resolver o hamiltoniano microscópico que modela um determinado sistema magnético de maneira a se obter sua equação de estado magnética: M(H,T). Para isto usamos o esquema de sub- redes adequado a diferentes arranjos magnéticos (ferro-, ferri- e antiferro- magnéticos). Com as soluções do hamiltoniano (autovalores e autovetores), grandezas físicas de interesse foram determinadas. Em princípio, consideramos sistemas com magnetismo localizado devido a elétrons 3d e 4f com participação de ligandos não magnéticos incluindo sistemas 3d-4f com presença de campo cristalino. Neste trabalho de dissertação empregamos o modelo de duas e três sub- redes para obter as equações de estado nos seguintes sistemas: troca tipo RKKY em RNi2B2C, supertroca em (Y3-zRz)(T1xFe1-x)(T2yFe3-y)O12, LixFe3-xO4 e (NixMn1- x)1.5[Cr(CN)6]. Nestas fórmulas, R representa um íon de terra rara, T1 e T2 representam íons não magnéticos. Alguns casos representativos são apresentados para exemplificar as diferentes equações de estado e comportamento das sub- redes, metamagnetismo, temperatura de compensação etc. Em princípio, a extensão para outros sistemas semelhantes, usando os modelos apresentados aqui, pode ser direta ou precisar incorporar parâmetros fenomenológicos adicionais.