Abstract In this paper, we study the nature of phase transition of the two-dimensional six-state Potts model under the external random magnetic field. The six-state Potts model exist temperature-dependent first-order phase transition. When the external random field is applied, the nature of phase can be altered from first-order to second-order.By employing the Monte Carlo simulation method, we inspected the energy histogram and Binder parameter of the six-state Potts model under the external random magnetic field. According to our analyses, the evidences reveal that the phase transition does not change until the external magnetic field is greater then 0.02

We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. We accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. The result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.

We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. We accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. The result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.

Depuis la proposition en 2006 d’utiliser des nano aimants réalisés par des techniques top-down pour reproduire des « spins artificiels », l’étude des systèmes de spins artificiels a suscité un large intérêt. En effet la possibilité de pouvoir réaliser arbitrairement tous types de réseaux de spins artificiels et de pouvoir imager les configurations magnétiques de ceux-ci dans l’espace direct, offre un large terrain de jeu dans le domaine de la physique statistique. Jusqu’à présent seuls des réseaux de spins d’Ising, multi axes (réseaux kagomé ou carré avec une aimantation planaire) ou plus récemment uni axes (avec une anisotropie perpendiculaire), ont été étudiés. Cependant en physique statistique d’autres modèles de spins sont étudiés et notamment les modèles de Potts à q-états. Au cours de cette thèse nous avons étudié le cas d’un modèle de Potts à 4 états, ayant la particularité de posséder uniquement des interactions dipolaires entre les spins: le modèle de Potts dipolaire. Nous avons tout d’abord réalisé une étude théorique, montrant que sur un réseau carré, en fonction de l’angle entre les spins et ce réseau, le système possède des états fondamentaux très différents : un ordre antiferromagnétique, un ordre respectant les règles de la glace (2 in- 2 out) ou un ordre ferromagnétique. Dans une deuxième partie, nous avons exposé l’étude expérimentale du modèle de Potts dipolaire. Des réseaux formés d’aimants carrés ayant 300 nm de côté ont été réalisés par lithographie électronique, à partir d’une couche épitaxiée de Fer possédant une anisotropie quadratique. A température ambiante, ces plots possèdent une configuration magnétique monodomaine pouvant prendre 4 directions équivalentes, comme recherché pour le modèle de Potts dipolaire à 4 états. Un passage à 350°C (inférieure à la température de Curie) sous champ nul permet d’activer thermiquement la réorientation des spins afin qu’ils se rapprochent de l’état fondamental de l’assemblée de spins. Les configurations magnétiques observées après recuit, à l’aide d’un microscope à force magnétique, montrent l’importance du couplage dipolaire sur les états obtenus, ainsi que l’influence de l’angle entre les spins et l’axe du réseau. Les différentes configurations prédites théoriquement sont bien observées / Since the proposal in 2006 to use nanomagnets patterned by top-down techniques to mimic "artificial spins", the studies of artificial spin systems has attracted wide interest. As a matter of facts, the possibility to design "upon request" arbitrary network and the possibility to determine completely the "spin" configuration with magnetic imaging offer a wide playground for statistical physics. Up to now only Ising spin systems, multi axes with planar magnetization (on square or Kagome lattice) or more recently, single axis with perpendicular anisotropy, have been studied. However, beyond Ising spins, statistical physics and condensed matter physics have shown the interest of other spin models like q-state Potts models. In this thesis, we introduce the dipolar 4-state Potts model. It is shown that on a square lattice, depending on the angle between spins and lattice, the system present very different properties like antiferromagnetic order, spin ice state (2 in-2 out ice rule) and even dipolar ferromagnetism. This model has been realized experimentally. 300 nm square magnets are patterned from a 2 nm thick Fe layer with cubic anisotropy. At room temperature, the magnets present a uniform state with 4 equivalent directions. Upon heating at 350 °C the magnets switch from one direction to another. It is therefore possible to simply drive the system toward its ground state. The magnetic configurations determined by magnetic force microscopy reveals the importance of the dipolar coupling as the different expected ground states (antiferromagnetic, spin ice and ferromagnetic) are indeed observed. It is noticeable that these very different properties are obtained with the same "spins" (magnetic elements) and same lattice

We study kinetics of phase segregation in multicomponent mixtures via Monte Carlo simulations of the q-state Potts model, in two spatial dimensions, for 2 ≤ q ≤ 20. The associated growth of domains in finite boxes, irrespective of q and temperature, can be described by a single universal finite-size scaling function, with only the introduction of a nonuniversal metric factor in the scaling variable. Our results show that although the scaling function is independent of the type of transition, the q-dependence of the metric factor hints to a crossover at q = 5 where the type of transition in the model changes from second to first order.

No description available.

The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures. The most studied model in statistical mechanics is thecelebrated Ising model. The Potts model is a natural extensionof the Ising model, and the beach model, which appears in adifferent mathematical context, is in certain respectsanalogous to the Ising model. The two main parts of this thesisdeal with the Potts model and the beach model,respectively. For theq-state Potts model on an infinite lattice, there areq+1 basic Gibbs measures: one wired-boundary measure foreach state and one free-boundary measure. For infinite trees,we construct "new" invariant Gibbs measures that are not convexcombinations of the basic measures above. To do this, we use anextended version of the random-cluster model together withcoupling techniques. Furthermore, we investigate the rootmagnetization as a function of the inverse temperature.Critical exponents to this function for different parametercombinations are computed. The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certain agreementpercolation event. We go on to study aq-state variant of the beach model. Using randomclustermodel methods again we obtain some results on where in theparameter space this model exhibits phase transition. Finallywe study the beach model on regular infinite trees as well.Critical values are estimated with iterative numerical methods.In different parameter regions we see indications of both firstand second order phase transition. Keywords and phrases:Potts model, beach model,percolation, randomcluster model, Gibbs measure, coupling,Markov chains on infinite trees, critical exponent.

A ênfase do trabalho recai na análise da taxa temporal de produção de entropia em sistemas de rede determinados por suas propriedades de simetria, no intuito de constituí-la como uma ferramenta paradigmática no estudo de sistemas irreversíveis. Nesse sentido, uma vez que se consegue consolidar uma definição consistente dessa grandeza para este campo, propõe-se uma abordagem estocástica para o modelamento da dinâmica dos exemplares considerados. O grupo de simetria C_ descreve as propriedades de invariância em sistemas com ampla relevância em física, de maneira que resulta natural elaborar a construção dos modelos em torno a sua estrutura, mais ainda quando se pode assumir que as características de simetria, no relativo à determinação da classe de universalidade dos modelos, são condições mais relevantes que a presença do não-equilíbrio, em coerência com a conjetura de Grinstein. Aliás, o modelo de Potts de três estados também apresenta propriedades de simetria próprias daquele grupo, além de ser passível de certo rigor no seu tratamento matemático, de maneira que oferece resultados consideravelmente satisfatórios e propícios para, por comparação, analisar os sistemas que concernem neste estudo. Assim, o procedimento tem como fim a determinação numérica da produção de entropia em sistemas com dinâmica irreversível e invariantes ante as transformações de simetria que compõem o grupo C_3v, partindo para isso de simulações de Monte Carlo em modelos estruturados sobre redes quadradas. A determinação da produção de entropia segue a prescrição de Schnakenberg (Schnakenberg [1976]), fundamentada nas correntes de probabilidade que surgem no sistema como consequência da violação da reversibilidade microscópica; a qual, por sua vez, estabelece a necessidade para os sistemas em equilíbrio de que todas as sequências cíclicas possíveis entre estados consecutivos sejam percorridas com igualdade de probabilidade num sentido quanto no inverso. Como uma segunda instância deste trabalho, também foram estudadas as propriedades de escalabilidade apresentadas por estos modelos durante seus primeiros instantes de evolução, isso, a partir da determinação numérica dos exponentes dinâmicos e sua caracterização dentro do marco teórico conhecido como \"short time scaling\'\' (Jansen et al. (1989)). Nesse sentido foram consideradas algumas das prescrições concernentes que tem apresentado melhor desempenho nos últimos anos. O observado mostra um comportamento coerente com a satisfação do que implicaria a conjetura de Grinstein para estes sistemas irreversíveis, indicando sua pertença à classe de universalidade do modelo de Potts de três estados, e, com isso, reafirmando os resultados obtidos em relação à produção de entropia. / This work is proposed as a study of the entropy production rate in lattice systems determined by its symmetries, looking for its consolidation as a paradigmatic tool in the area of irreversible and nonequilibrium systems. Henceforth, given the actual possibility of defining this quantity on that field, a stochastic perspective is adopted for modeling the dynamics of the considered systems. The C_ symmetry group describes the invariance properties of a wide range of physical systems, hence it results sensitive building the models to be dealt in accordance with its intrinsic characteristics, even more when it comes to be just fair, at the sight of the previous available analysis as well of the Grinstein conjecture, considering the invariance properties of systems as a more relevant factor than the reversibility conditions in what concerns the establishment of its universality class. The three states Potts model, indeed, shares its symmetry characteristics with those owned by the elements of the referred group and, also, concerning it there is a considerable amount of well confirmed information, becoming suitable for contrasting the results obtained. Henceforth, this work is focused on the numerical determination of the entropy production rate in irreversible systems whose invariance properties are the ones defined by the C_ symmetry group, implementing for that porpoise Monte Carlo simulations over square lattices models with the proper symmetries. By its own, the entropy production is determined in accordance with the Schnakenberg prescription (Scnhakenberg [1976]); deeply related with the probability currents emerging within irreversible systems as consequence of microscopic reversibility violation, which, in equilibrium, is imposed due to the mandatory equality between the evolution directions of all possible cyclic paths through a succession of states assumed by the system. As a second instance of this work, the scaling properties of the studied models during the first period of its evolution, just after the microscopic scale of time, were also analyzed. Henceforth, the determination of its dynamical exponents, as well as its characterization within the context of \"short time scaling\'\' (Jansent et al. [1989]) was realized through the calculus of some quantities with proven signatures of presenting an scalable \"initial slip\'\', finding strong suggestions for the models of being in the same universality class of the three state Potts model, fact coherent with the Grinstein conjecture if extended over them, but also with the observed behavior of the entropy production.

Made available in DSpace on 2015-03-05T13:56:59Z (GMT). No. of bitstreams: 0 Previous issue date: 10 / Hewlett-Packard Brasil Ltda / A simulação de sistemas celulares tem recebido grande interesse nos últimos anos. Em particular, o modelo de Potts celular é o mais utilizado na área dada a sua precisão em representar estes sistemas. Este modelo, na sua forma convencional, possui uma série de operações e cálculos que são executados de maneira pouco eficiente, o que impossibilita sua utilização em simulações grandes e que exigem considerável tempo e memória para sua conclusão. Com base nisso propomos um novo algoritmo de maior desempenho que permite obter resultados aproximados dos obtidos com o algoritmo Monte Carlo em tempo bem menor. Técnicas de execução concorrente e comunicação foram introduzidas no algoritmo através do uso de processos leves para execução em computadores com memória compartilhada e usando aglomerados de computadores, respectivamente, buscando reduzir o tempo de processamento e viabilizando a execução de simulações de grande porte. Os resultados obtidos de simulações de segregação celular e evolução de espumas mostra / The simulation of cellular systems has received great interest in the last years. In particular, the cellular Potts model is widely used in the area given its precision in representing these systems. This model, in its standard form, takes a series of operations and calculations that are executed in an ine±cient way, what disables its use in large scale simulations that demand considerable time and memory for conclusion. Based on that, we propose a new algorithm of higher performance that allows to obtain results close from those obtained with the Monte Carlo algorithm in much shorter time. Techniques of concurrent execution and communication have been introduced in the algorithm, through the use of light processes for execution in computers with shared memory and using clusters of computers, respectively, aiming to reduce the processing time and making possible the execution of large scale simulations. The results presented obtained from simulation of cellular segregation and foam evolution show a minimum s

Neste trabalho estudamos o comportamento crítico do modelo de Potts p-estados na árvore de Cayley, através das propriedades do conjunto de zeros de Yang-Lee da função de partição. Tratando a transformação do grupo de renormalização como um mapeamento racional na esfera de Riemann utiliza-se alguns resultados da teoria de Julia e Fatou para obter-se uma descrição geométrica do comportamento crítico do modelo. Mostra-se de que forma o conjunto de zeros de Yang-Lee se relaciona com o conjunto de Julia do mapa do grupo de renormalização, e calculam-se alguns parâmetros geométricos desse conjunto que descrevem o comportamento não universal do modelo. / We study the critical behavior of the p-state Potts model on a Cayley tree, looking for the properties of the Yang-Lee zeros set of the partition function. We treated the renormalization group transformation as a rational mapping on the Riemann sphere, and use some results from the Julia and Fatou theory to obtain a geometrical description of the critical properties of the model. We show how the Yang-Lee zeros set is associated with the Julia set of the renormalization group map, and we also calculate some geometrical parameters of this set which describes the non-universal behavior of the model.