Thesis (MSc (Physics))--Stellenbosch University, 2008. / Nuclear physics can be applied in various ways to the study of neutron stars. This thesis reports on one such application, where the relativistic mean-field approximation has been employed to calculate the equations of state of matter in the neutron star interior. In particular the equations of state of nuclear and neutron star matter of the NL3, PK1 and FSUGold parameter sets were derived. A survey of available literature on neutron stars is presented and we use the derived equations of state to reproduce the properties of saturated nuclear matter as well as the mass-radius relationship of a static, spherical symmetric neutron star. Results are compared to published values of the properties of saturated nuclear matter and to available observational data of the mass-radius relationship of neutron stars.

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L'objectif de ce travail est de prédire les évolutions microstructurales se produisant dans le tantale pur lors d'un traitement thermique en fonction de son état microstructural initial. La restauration, la recristallisation et la croissance de grains sont décrites à l'aide d'un modèle en champ moyen qui nécessite une description adéquate de la microstructure, en termes de distributions de tailles de grains et de densités de dislocations équivalentes. La densité de dislocation équivalente moyenne peut être évaluée par une simple mesure de dureté Vickers. L'établissement de la relation dureté-densité de dislocations nécessite l'utilisation d'une loi de comportement basée sur la densité de dislocations équivalente. Les évolutions microstructurales au cours d'un traitement thermique ont été observées et les paramètres pilotant ces phénomènes ont été identifiés à l'aide d'essais originaux comme l'observation in situ de la recristallisation ou l'utilisation d'essais à gradient de déformation pour déterminer le seuil de densité de dislocations équivalente pour déclencher la recristallisation. Des essais plus classiques ont permis d'obtenir des cinétiques de recristallisation dans la gamme 1000°C-1100°C pour différentes microstructures initiales. Les simulations des différents traitements thermiques à l'aide du modèle à champ moyen rendent bien compte des évolutions microstructurales en termes de fraction recristallisée et de taille des grains recristallisés pour des microstructures faiblement déformées ou fortement déformées et fragmentées, en utilisant une description adéquate du type de microstructure initiale. Le modèle devra en revanche être adapté pour traiter le cas de microstructures intermédiaires, en enrichissant non seulement la description de la microstructure initiale mais également celle de l'étape de germination des grains recristallisés. Il deviendra alors capable de prédire les évolutions de microstructures pour tout type de microstructure initiale de tantale. / This study aims at predicting the microstructural evolution of pure tantalum during annealing according the initial microstructural state. Static recovery and discontinuous recrystallization as well as grain growth are described using a mean-field model requiring an appropriate description of the microstructure, using both equivalent dislocation densities and grain sizes distributions. The average equivalent dislocation density can be assessed from Vickers microhardness measurements. The calibration of such a relation between microhardness and dislocation density involves the use of a dislocation density-based constitutive law. Microstructural evolutions during annealing have been observed and control parameters of these phenomena have been determined using original tests such as in situ observation of the recrystallization process or the use of strain gradient samples to assess the critical dislocation density for the onset of recrystallization. More classical tests have been carried out to get recrystallization kinetics in the range 1000-1100°C for different initial microstructures. Simulations of annealing using the mean-field model adapted for tantalum match the experimental evolution of both recrystallized fraction and recrystallized grain size, in either weakly deformed or severely deformed and fragmented microstructures. On the other hand, the model needs to be further adapted for intermediate microstructures, with both a more elaborate description of the initial microstructure and of the nucleation stage of the recrystallized grains. It will then be suitable to predict evolutions of any initial tantalum microstructure during annealing.

Dans cette thèse, nous étudions des réseaux sans fil dans lesquels les terminaux mobiles sont autonomes dans le choix de leurs configurations de communication. Cette autonomie de décision peut notamment concerner le choix de la technologie d'accès au réseau, le choix du point d'accès, la modulation du signal, les bandes de fréquences occupées, la puissance du signal émis, etc. Typiquement, ces choix de configuration sont réalisés dans le but de maximiser des métriques de performances propres à chaque terminal. Sous l'hypothèse que les terminaux prennent leurs décisions de manière rationnelle afin de maximiser leurs performances, la théorie des jeux s'applique naturellement pour modéliser les interactions entre les décisions des différents terminaux. Plus précisément, l'objectif principal de cette thèse est d'étudier des stratégies d'équilibre de contrôle de puissance d'émission afin de satisfaire des considérations d'efficacité énergétique. Le cadre des jeux stochastiques est particulièrement adapté à ce problème et nous permet notamment de caractériser la région de performance atteignable pour toutes les stratégies de contrôle de puissance qui mènent à un état d'équilibre. Lorsque le nombre de terminaux en jeu est grand, nous faisons appel à la théorie des jeux à champ moyen pour simplifier l'étude du système. Cette théorie nous permet d'étudier non pas les interactions individuelles entre les terminaux, mais l'interaction de chaque terminal avec un champ moyen qui représente l'état global des autres terminaux. Des stratégies de contrôle de puissance optimales du jeu à champ moyen sont étudiées. Une autre partie de la thèse a été consacrée à des problématiques d'apprentissage de points d'équilibre dans les réseaux distribués. En particulier, après avoir caractérisé les positions d'équilibre d'un jeu de positionnement de points d'accès, nous montrons comment des dynamiques de meilleures réponses et d'apprentissage permettent de converger vers un équilibre. Enfin, pour un jeu de contrôle de puissance, la convergence des dynamiques de meilleures réponses vers des points d'équilibre a été étudiée. Il est notamment proposé un algorithme d'adaptation de puissance convergeant vers un équilibre avec une faible connaissance du réseau. / In this thesis, we study wireless networks in which mobile terminals are free to choose their communication configuration. Theses configuration choices include access wireless technology, access point association, coding-modulation scheme, occupied bandwidth, power allocation, etc. Typically, these configuration choices are made to maximize some performance metrics associated to every terminals. Under the assumption that mobile terminals take their decisions in a rational manner, game theory can be applied to model the interactions between the terminals. Precisely, the main objective of this thesis is to study energy-efficient power control policies from which no terminal has an interest to deviate. The framework of stochastic games is particularly suited to this problem and allows to characterize the achievable utility region for equilibrium power control strategies. When the number of terminals in the network is large, we invoke mean field game theory to simplify the study of the system. Indeed, in a mean field game, the interactions between a player and all the other players are not considered individually. Instead, one only studies the interactions between each player and a mean field, which is the distribution of the states of all the other players. Optimal power control strategies from the mean field formulation are studied. Another part of this thesis has been focused on learning equilibria in distributed games. In particular, we show how best response dynamics and learning algorithms can converge to an equilibrium in a base station location game. For another scenario, namely a power control problem, we study the convergence of the best response dynamics. In this case, we propose a power control behavioral rule that converges to an equilibrium with very little information about the network.

Os processos de negócio desempenham um papel muito importante na indústria, principalmente pela evolução das tecnologias da informação. As plataformas de computação em nuvem, por exemplo, com a alocação de recursos computacionais sob demanda, possibilitam a execução de processos altamente requisitados. Para tanto, é necessário definir o ambiente de execução dos processos de tal modo que os recursos sejam utilizados de forma ótima e seja garantida a correta funcionalidade do processo. Nesse contexto, diferentes métodos já foram propostos para modelar os processos de negócio e analisar suas propriedades quantitativas e qualitativas. Há, contudo, vários desafios que podem restringir a aplicação desses métodos, especialmente para processos com alta demanda (como os workflows de numerosas instâncias) e que dependem de recursos limitados. A análise de desempenho de workflows de numerosas instâncias via modelagem analítica é o objeto de estudo deste trabalho. Geralmente, para a realização desse tipo de análise usa-se modelos matemáticos baseados em técnicas Markovianas (sistemas estocásticos), que sofrem do problema da explosão do espaço de estados. Entretanto, a Teoria do Campo Médio indica que o comportamento de um sistema estocástico, sob certas condições, pode ser aproximado por o de um sistema determinístico, evitando a explosão do espaço de estados. Neste trabalho usamos tal estratégia e, com base na definição formal de aproximação determinística e suas condições de existência, elaboramos um método para representar os workflows, e seus recursos, como equações diferenciais ordinárias, que descrevem um sistema determinístico. Uma vez definida a aproximação determinística, realizamos a análise de desempenho no modelo determinístico, verificando que os resultados obtidos são uma boa aproximação para a solução estocástica. / Business processes play a very important role in the industry, especially by the evolution of information technologies. Cloud computing platforms, for example, with the allocation of on-demand computing resources enable the execution of highly requested processes. Therefore, it is necessary to define the execution environment of the processes in such a way that the resources are used optimally and the correct functionality of the process is guaranteed. In this context, different methods have already been proposed to model business processes and analyze their quantitative and qualitative properties. There are, however, a number of challenges that may restrict the application of these methods, especially for high-demanded processes (such as workflows of numerous instances) and that rely on resources that are limited. The analysis of the performance of workflows of numerous instances through analytical modeling is the object of study of this work. Generally, for the accomplishment of this type of analysis, mathematical models based on Markovian techniques (stochastic systems) are used, which suffer the problem of the state space explosion. However, the Mean Field Theory, indicates that the behavior of a stochastic system, under certain conditions, can be approximated by that of a deterministic system, avoiding the explosion of the state space. In this work we use such a strategy, based on the formal definition of deterministic approximation and its conditions of existence, we elaborate a method to represent the workflows, and their resources, as ordinary differential equations, which describe a deterministic system. Once the deterministic approximation has been defined, we perform the performance analysis in the deterministic model, verifying that the obtained results are a good approximation for the stochastic solution.

Estudamos sistemas de passeios aleatórios sobre os vértices de um grafo completo. Inicialmente há uma partícula em cada vértice do grafo das quais somente uma está ativa, as outras estão inativas. A partícula ativa realiza um passeio aleatório simples a tempo discreto com tempo de vida que depende do passado do processo, movendo-se ao longo de elos. Quando uma partícula ativa encontra uma inativa, esta se ativa; quando salta sobre um vértice já visitado, morre. O objetivo desta dissertação é estudar a cobertura do grafo completo, ou seja, a proporção de vértices visitados ao fim do processo, quando o número $n$ de vértices tende ao infinito. Analisamos as equações de campo médio para o processo descrito acima, comparando os seus resultados com os do modelo aleatório. Aqui, os resultados do campo médio parecem reproduzir os do modelo aleatório. Depois, apresentamos um estudo similar entre o modelo estocástico e as equações de campo médio para o caso em que cada partícula possui 2 vidas. Finalmente, observamos a cobertura do grafo completo para as equações de campo médio quando o número de vidas por partículas é maior que dois. / We study random walks systems on complete graphs. Initially there is a particle at each vertex of the graph; only one is active and the other are inactive. An active particle performs a discrete-time simple random walk with lifetime depending on the past of the process moving along edges. When an active particle hits an inactive one, the latter is activated. When it jumps on a vertex which has been visited before it dies. The goal of this work is to study the coverage of the complete graph, that is, the proportion of visited vertices at the end of the process, when the number of vertices goes to infinity. We analyze the mean field equations to the process cited above, comparing their results with the ones of the random model. Here the results of the mean field approach seem to reproduce the ones of the random model. After we present a similar study between the stochastic model and mean field approximation to the case that each particle has 2 lifes. Finally we observe the coverage of the complete graph to the mean-field equations when the number of lifes by particle is bigger than two.

Estudamos sistemas de passeios aleatórios sobre os vértices de um grafo completo. Inicialmente há uma partícula em cada vértice do grafo das quais somente uma está ativa, as outras estão inativas. A partícula ativa realiza um passeio aleatório simples a tempo discreto com tempo de vida que depende do passado do processo, movendo-se ao longo de elos. Quando uma partícula ativa encontra uma inativa, esta se ativa; quando salta sobre um vértice já visitado, morre. O objetivo desta dissertação é estudar a cobertura do grafo completo, ou seja, a proporção de vértices visitados ao fim do processo, quando o número $n$ de vértices tende ao infinito. Analisamos as equações de campo médio para o processo descrito acima, comparando os seus resultados com os do modelo aleatório. Aqui, os resultados do campo médio parecem reproduzir os do modelo aleatório. Depois, apresentamos um estudo similar entre o modelo estocástico e as equações de campo médio para o caso em que cada partícula possui 2 vidas. Finalmente, observamos a cobertura do grafo completo para as equações de campo médio quando o número de vidas por partículas é maior que dois. / We study random walks systems on complete graphs. Initially there is a particle at each vertex of the graph; only one is active and the other are inactive. An active particle performs a discrete-time simple random walk with lifetime depending on the past of the process moving along edges. When an active particle hits an inactive one, the latter is activated. When it jumps on a vertex which has been visited before it dies. The goal of this work is to study the coverage of the complete graph, that is, the proportion of visited vertices at the end of the process, when the number of vertices goes to infinity. We analyze the mean field equations to the process cited above, comparing their results with the ones of the random model. Here the results of the mean field approach seem to reproduce the ones of the random model. After we present a similar study between the stochastic model and mean field approximation to the case that each particle has 2 lifes. Finally we observe the coverage of the complete graph to the mean-field equations when the number of lifes by particle is bigger than two.

Beaucoup de systèmes qui ont un certain degré de désordre ont des similaritésdans leur structure: le paysage énergétique est aléatoire et il a plusieursminima locaux de l’énergie. Quand on ajoute une petite perturbation externeau système à basse temprature, il est raisonnable d’attendre que la dynamiqueconduira le système d’un minimum à l’autre, et ça donne lieu à une réponsealéatoire et saccadé. Les sautes discontinus que l’on observe sont appelésavalanches, et l’intérêt de ce travail est le calcul de leur distribution. Undes résultats est en effet le développement d’un cadre pour calculer cettedistribution dans des systèmes en dimension infinie qui peuvent être décritsavec le replica symmetry breaking. Nous appliquons les résultats à l’un desmodèles les plus simples des verres structuraux, c’est à dire les empilementsdenses de sphères molles avec répulsion harmonique, avec une déformation(shear strain) du volume comme perturbation. Nous soutenons que, quandla déformation est suffisamment petite, une portion de la distribution desavalanches devient une loi de puissance, dont l’exposant peut être directementlié au paramètre d’ordre de la brisure de symétrie de replica. Cet exposant estégalement lié à la distribution des forces de contact (au moins entre certainessphères), dont le comportement asymptotique on sais que ne dpends pasfortement de la dimension spatiale; pour cette raison nous comparons lesprdictions de champ moyen en dimension infinie avec des simulation du mêmesystème en dimension trois et, remarquablement, on trouve un bon accord.Dans le reste de la thèse nous discutons aussi les similarités avec des travauxprécédents et quelques consquences que la distribution des avalanches donnesur les propriétés élastiques de la matière granulaire dense. / Many systems that are somehow characterized by a degree of disorder sharea similar structure: the energy landscape has many sample-dependent localenergy minima. When a small external perturbation is applied to the systemat low temperature, it is reasonable to expect that the dynamics will leadthe system from a minimum to another, thus displaying a random and jerkyresponse. The discontinuous jumps that one observes are called avalanches,and the focus of this work is the computation of their distribution. Oneof the results is indeed the development of a framework that allows thecomputation of this distribution in infinite-dimensional systems that canbe described within a replica symmetry breaking ansatz. We apply theresults to one of the simplest models of structural glasses, namely densepackings of (harmonic) soft spheres, either at jamming or at larger densities,subject to a shear transformation that induces jumps both in the totalenergy and in the shear stress of the system. We argue that, when theshear strain is small enough, the avalanche distribution develops a power-lawbehavior, whose exponent can be directly related to the functional orderparameter of the replica symmetry breaking solution. This exponent is alsorelated to the distribution of contact forces (or at least of the contact forcesbetween some of the spheres), whose asymptotic behavior is known not todepend strongly on the spatial dimension; for this reason, we compare theinfinite-dimensional prediction with three dimensional simulations of thesame systems and, remarkably, we find a good agreement. In the rest of thethesis we compare our results with previous works, and we also discuss someof the consequences that the avalanche distribution lead to, concerning thestatistical elastic properties of dense granular media.

Spinodal instability mechanism and early development of density fluctuations for symmetric nuclear matter at finite temperature are studied. A stochastic extension of Walecka-type relativistic mean-field model including non-linear self-interactions of scalar mesons with NL3 parameter set is employed in the semi-classical approximation. The growth rates of unstable collective modes are investigated below the normal density and at low temperatures. The system exhibits most unstable behavior in longer wave lengths at baryon densities &rho / B = 0.4 &rho / 0 , while most unstable behavior occurs in shorter wavelengths at lower baryon densities &rho / B = 0.2 &rho / 0 . The unstable response of the system shifts towards longer wavelengths with the increasing temperature at both densities. The early growth of the density correlation functions are calculated, which provide valuable information about the initial size of the condensation and the average speed of condensing fragments. Furthermore, the relativistic results are compared with Skyrme type non-relativistic calculations. Qualitatively similar results are found in both non-relativistic and relativistic descriptions.

In this thesis we employ various mean-field approaches to study the shortrange interaction effects in topological insulators. We start with the Kane-Mele model on the decorated honeycomb lattice and study the stability of topological insulator phase against different perturbations. We establish an adiabatic connection between a noninteracting topological insulator and a strongly interacting spin liquid in its Majorana fermion representation. We use the Hartree-Fock mean-field approach, slave-rotor approach and slave-boson approach to study correlation effects related to topological insulators. With the spontaneous symmetry breaking mechanism, we can have an interaction driven topological insulator with extended Hubbard models on the kagome lattice and decorated honeycomb lattice. For the interplay among spin-orbit coupling, distortion and correlation effect in transition metal oxides, we use the slave-rotor mean-field approach to study its phase transition. We identify regimes where a strong topological Mott insulator and a weak topological insulator reside due to the strong Coulomb interaction and distortion. This is relevant to experiments with the transition metal oxides as they hold promise to realize topological insulators. To study the doping effects and a possible spin liquid in Kane-Mele-Hubbard model on the honeycomb lattice, we employ the slave-boson mean-field approach which is appropriate for the intermediate interaction strength. We compare our results with those obtained from other methods. / text