Encontrar um modo de criar imagens fotorealísticas tem sido uma meta da Computação Gráfica por muitos anos [GLA 89]. Neste sentido, os aspectos que possuem principal importância são a modelagem e a iluminação. Ao considerar aspectos de modelagem, a obtenção de realismo mostra-se bastante difícil quando se pretende, através de técnicas tradicionais de modelagem, modelar objetos cujas formas não são bem definidas. Dentre alguns exemplos destes tipos de objetos, podem-se citar fogo, fumaça, nuvens, água, etc. Partindo deste fato, Reeves [REE 83] introduziu uma técnica denominada sistemas de partículas para efetuar a modelagem de fogo e explosões. Um sistema de partículas pode ser visto como um conjunto de partículas que evoluem ao longo do tempo. Os procedimentos envolvidos na animação de um sistema de partículas são bastante simples. Basicamente, a cada instante de tempo, novas partículas são geradas, os atributos das partículas antigas são alterados, ou estas partículas podem ser extintas de acordo com certas regras pré-definidas. Como as partículas de um sistema são entidades dinâmicas, os sistemas de partículas são especialmente adequados para o uso em animação. Ainda, dentre as principais vantagens dos sistemas de partículas quando comparados com as técnicas tradicionais de modelagem, podem-se citar a facilidade da obtenção de efeitos sobre as partículas (como borrão de movimento), a necessidade de poucos dados para a modelagem global do fenômeno, o controle por processos estocásticos, o nível de detalhamento ajustável e a possibilidade de grande controle sobre as suas deformações. Entretanto, os sistemas de partículas possuem algumas limitações e restrições que provocaram o pouco desenvolvimento de algoritmos específicos nesta área. Dentre estas limitações, as principais são a dificuldade de obtenção de efeitos realísticos de sombra e reflexão, o alto consumo de memória e o fato dos sistemas de partículas possuírem um processo de animação específico para cada efeito que se quer modelar. Poucos trabalhos foram desenvolvidos especificamente para a solução destes problemas, sendo que a maioria se destina à modelagem de fenômenos através de sistemas de partículas. Tendo em vista tais deficiências, este trabalho apresenta métodos para as soluções destes problemas. É apresentado um método para tornar viável a integração de sistemas de partículas em ambientes de Ray Tracing, através do uso de uma grade tridimensional. Também, são apresentadas técnicas para a eliminação de efeitos de aliasing das partículas, assim como para a redução da quantidades de memória exigida para o armazenamento dos sistemas de partículas. Considerando aspectos de animação de sistemas de partículas, também é apresentado uma técnica de aceleração para a detecção de colisões entre o sistema de partículas e os objetos de uma cena, baseada no uso de uma grade pentadimensional. Aspectos relativos à implementação, tempo de processamento e fatores de aceleração são apresentados no final do trabalho, assim como as possíveis extensões futuras e trabalhos sendo realizados. / Finding a way to create photorealistic images has been a goal of Computer Graphics for many years [GLA 89]. In this sense, the aspects that have main importance are modeling and illumination. Considering aspects of modeling, the obtention of realism is very difficult when it is intended to model fuzzy objects using traditional modeling techniques. Among some examples of these types of objects, fire, smoke, clouds, water, etc. can be mentioned. With this fact in mind, Reeves [REE 83] introduced a technique named particle systems for modeling of fire and explosions. A particle system can be seen as a set of particles that evolves over time. The procedures involved in the animation of particle systems are very simple. Basically, at each time instant, new particles are generated, the attributes of the old ones are changed, or these particles can be extinguished according to predefined rules. As the particles of a system are dynamic entities, particle systems are specially suitable for use in animation. Among the main advantages of particle systems, when compared to traditional techniques, it can be mentioned the facility of obtaining effects such as motion blur over the particles, the need of few data to the global modeling of a phenomen, the control by stochastic processes, an adjustable level of detail and a great control over their deformations. However, particle systems present some limitations and restrictions that cause the little development of specific algorithms in this area. Among this limitations, the main are the difficulty of obtention of realistic effects of shadow and reflection, the high requirement of memory and the fact that particle systems need a specific animation process for each effect intended to be modeled. Few works have been developed specifically for the solution of these problems; most of them are developed for the modeling of phenomena through the use of particle systems. Keeping these deficiencies in mind, this work presents methods for solving these problems. A method is presented to make practicable the integration of particle systems and ray tracing, through the use of a third-dimensional grid. Also, a technique is presented to eliminate effects of aliasing of particles, and to reduce the amount of memory required for the storage of particle systems. Considering particle systems animation, a technique is also presented to accelerate the collision detection between particle systems and the objects of a scene, based on the use of a fifth-dimensional grid. Aspects related to the implementation, processing time and acceleration factors are presented at the end of the work, as well as the possible future extensions and ongoing works.

Le processus d’exclusion simple est défini formellement de la façon suivante : chaque particule effectue une marche aléatoire sur un ensemble de sites et interagit avec les autres particules en ne se déplaçant jamais sur un site occupé.Malgré sa simplicité, ce processus présente des propriétés que l’on retrouve dans beaucoup de modèles de mécanique statistique plus complexes. C’est la conjonction de la simplicité du processus et de l’intérêt des phénomènes observés quien fait l’un des modèles de référence en mécanique statistique hors équilibre. Dans cette thèse, je me suis intéressé au cas du processus d’exclusion totalement asymétrique (les particules sautent uniquement vers la droite) sur Nafin d’étudier son comportement en fonction du mécanisme de création de particules: on crée des particules au site 0 avec un taux dépendant de la configurationactuelle. Dès que ce mécanisme n’est plus un processus de Poisson, le processusd’exclusion associé n’admet plus de mesure invariante sous forme de produitce qui fait que les méthodes classiques de calcul sur le générateur infinitésimaln’aboutissent que rarement. Je me suis donc appuyé principalement sur les méthodesde couplage et de particules de deuxième classe.Dans la première partie de la thèse, je me suis intéressé au modèle introduitpar Grosskinsky pour lequel j’ai obtenu les résultats suivants : si le taux maximumde création et la densité initiale de particules sont plus petits que 12 et sile mécanisme de création est à portée intégrable, il n’y a pas de transition dephase c’est-à-dire qu’il n’y a qu’une seule mesure invariante.Dans la deuxième partie de la thèse, je me suis intéressé au problème inversedont le but est de construire un processus à portée finie mais non-intégrableayant une transition de phase. Pour cela, je me suis inspiré des méthodes développéespour le processus des spécifications de Bramson et Kalikow. / The simple exclusion process is formally defined as follows : each particle performs a simple random walk on a set of sites and interacts with other particles by never moving on occupied sites. Despite its simplicity, this process has properties that are found in many more complex statistical mechanics models. It is the combination of the simplicity of the process and the importance of the observed phenomena that make it one of the reference models in out of equilibrium statistical mechanics. In this thesis, I’m interested in the case of the totally asymmetric exclusion process (particles jump only to the right) on N to study its behavior according to the mechanism of particle creation : particles are created at site 0 with arate depending on the current configuration. Once this mechanism is no longer a Poisson process, the associated exclusion process does not admit a product invariant measure. As a consequence, classical computation methods with theinfinitesimal generator are rarely successful. So I used mainly the methods of coupling and second class particles.In the first part of the thesis, I’m interested in the model introduced by Grosskinsky for which I get the following result : if the maximum rate of creation and the initial density of particles are smaller than 12 and if the creation mechanism is of integrable range, there is no phase transition which means that there is only one invariant measure. In the second part of the thesis, my goal was to construct a process with finite and non-integrable range that has a phase transition. For this, I was inspired by methods developed for the process of specification of Bramson and Kalikow.

Em simulações físicas baseadas em partículas, a informação sobre quais partículas pertencem à fronteira do sistema e quais são consideradas internas é, em geral, uma informação útil porém difícil de ser obtida eficientemente. Esta informação pode ser usada na geração da superfície livre de um fluido ou no cálculo da tensão superficial o mesmo, entre outras aplicações. Técnicas encontradas na literatura podem apresentar resultados satisfatórios, mas em geral são sensíveis à escala do problema, distribuição das partículas e envolvem operações computacionalmente caras como inversão de matrizes. O objetivo deste trabalho é estudar os métodos existentes e apresentar uma alternativa com custo computacional mais baixo e que seja capaz de lidar com problemas de diferentes escalas e naturezas de forma mais simples que os métodos existentes. / In particle-based physics simulations, the information about which particles belong to the boundary of the system and which are considered internal is, in general, an information that is useful but hard to obtain in an efficient way. This information can be applied to generate the free surface of the fluid or to compute the surface tension, among other applications. Techniques found in the literature may present satisfactory results, but in general they are sensible to the problem scale, particle distribution and involve computationally expensive operations such as matrix inversion. The goal of this study is to analyze the existing methods and present an alternative with lower computational cost and which is capable of handling problems with different scales and nature in a simpler way than the existing methods.

Encontrar um modo de criar imagens fotorealísticas tem sido uma meta da Computação Gráfica por muitos anos [GLA 89]. Neste sentido, os aspectos que possuem principal importância são a modelagem e a iluminação. Ao considerar aspectos de modelagem, a obtenção de realismo mostra-se bastante difícil quando se pretende, através de técnicas tradicionais de modelagem, modelar objetos cujas formas não são bem definidas. Dentre alguns exemplos destes tipos de objetos, podem-se citar fogo, fumaça, nuvens, água, etc. Partindo deste fato, Reeves [REE 83] introduziu uma técnica denominada sistemas de partículas para efetuar a modelagem de fogo e explosões. Um sistema de partículas pode ser visto como um conjunto de partículas que evoluem ao longo do tempo. Os procedimentos envolvidos na animação de um sistema de partículas são bastante simples. Basicamente, a cada instante de tempo, novas partículas são geradas, os atributos das partículas antigas são alterados, ou estas partículas podem ser extintas de acordo com certas regras pré-definidas. Como as partículas de um sistema são entidades dinâmicas, os sistemas de partículas são especialmente adequados para o uso em animação. Ainda, dentre as principais vantagens dos sistemas de partículas quando comparados com as técnicas tradicionais de modelagem, podem-se citar a facilidade da obtenção de efeitos sobre as partículas (como borrão de movimento), a necessidade de poucos dados para a modelagem global do fenômeno, o controle por processos estocásticos, o nível de detalhamento ajustável e a possibilidade de grande controle sobre as suas deformações. Entretanto, os sistemas de partículas possuem algumas limitações e restrições que provocaram o pouco desenvolvimento de algoritmos específicos nesta área. Dentre estas limitações, as principais são a dificuldade de obtenção de efeitos realísticos de sombra e reflexão, o alto consumo de memória e o fato dos sistemas de partículas possuírem um processo de animação específico para cada efeito que se quer modelar. Poucos trabalhos foram desenvolvidos especificamente para a solução destes problemas, sendo que a maioria se destina à modelagem de fenômenos através de sistemas de partículas. Tendo em vista tais deficiências, este trabalho apresenta métodos para as soluções destes problemas. É apresentado um método para tornar viável a integração de sistemas de partículas em ambientes de Ray Tracing, através do uso de uma grade tridimensional. Também, são apresentadas técnicas para a eliminação de efeitos de aliasing das partículas, assim como para a redução da quantidades de memória exigida para o armazenamento dos sistemas de partículas. Considerando aspectos de animação de sistemas de partículas, também é apresentado uma técnica de aceleração para a detecção de colisões entre o sistema de partículas e os objetos de uma cena, baseada no uso de uma grade pentadimensional. Aspectos relativos à implementação, tempo de processamento e fatores de aceleração são apresentados no final do trabalho, assim como as possíveis extensões futuras e trabalhos sendo realizados. / Finding a way to create photorealistic images has been a goal of Computer Graphics for many years [GLA 89]. In this sense, the aspects that have main importance are modeling and illumination. Considering aspects of modeling, the obtention of realism is very difficult when it is intended to model fuzzy objects using traditional modeling techniques. Among some examples of these types of objects, fire, smoke, clouds, water, etc. can be mentioned. With this fact in mind, Reeves [REE 83] introduced a technique named particle systems for modeling of fire and explosions. A particle system can be seen as a set of particles that evolves over time. The procedures involved in the animation of particle systems are very simple. Basically, at each time instant, new particles are generated, the attributes of the old ones are changed, or these particles can be extinguished according to predefined rules. As the particles of a system are dynamic entities, particle systems are specially suitable for use in animation. Among the main advantages of particle systems, when compared to traditional techniques, it can be mentioned the facility of obtaining effects such as motion blur over the particles, the need of few data to the global modeling of a phenomen, the control by stochastic processes, an adjustable level of detail and a great control over their deformations. However, particle systems present some limitations and restrictions that cause the little development of specific algorithms in this area. Among this limitations, the main are the difficulty of obtention of realistic effects of shadow and reflection, the high requirement of memory and the fact that particle systems need a specific animation process for each effect intended to be modeled. Few works have been developed specifically for the solution of these problems; most of them are developed for the modeling of phenomena through the use of particle systems. Keeping these deficiencies in mind, this work presents methods for solving these problems. A method is presented to make practicable the integration of particle systems and ray tracing, through the use of a third-dimensional grid. Also, a technique is presented to eliminate effects of aliasing of particles, and to reduce the amount of memory required for the storage of particle systems. Considering particle systems animation, a technique is also presented to accelerate the collision detection between particle systems and the objects of a scene, based on the use of a fifth-dimensional grid. Aspects related to the implementation, processing time and acceleration factors are presented at the end of the work, as well as the possible future extensions and ongoing works.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / AnimaÃÃo baseada em FÃsica ganhou muita atenÃÃo dos pesquisadores nas Ãltimas duas dÃcadas, por causa dos desafios de realismo e desempenho propostos pelas indÃstrias de filme e jogos, bem como pelas aplicaÃÃes de realidade virtual e comÃrcio eletrÃnico. Consequentemente, foram feitos avanÃos significativos nessa Ãrea, e vÃrias abordagens foram propostas, tanto para a modelagem de tecidos, quanto para a simulaÃÃo de seu comportamento dinÃmico. Modelar tecido como um sistema massa-mola tem sido uma tÃcnica amplamente utilizada, por causa de sua simplicidade, e porque ela permite soluÃÃes numÃricas mais eficientes para as equaÃÃes diferenciais que governam o problema. Esses mÃtodos de integraÃÃo sÃo divididos em duas categorias principais: os mÃtodos explÃcitos e os mÃtodos implÃcitos. Um conhecimento mais aprofundado sobre o uso desses mÃtodos no contexto de simulaÃÃo dinÃmica de tecidos ainda à pertinente. Dessa forma, o objetivo desta dissertaÃÃo à analisar os mÃtodos de integraÃÃo mais comumente usados na literatura sobre simulaÃÃo de tecidos baseados em sistemas de partÃculas, para entender melhor suas vantagens e desvantagens. Um estudo comparativo desses mÃtodos à feito, considerando os critÃrios de realismo de animaÃÃo, esforÃo computacional e estabilidade numÃrica. / Physically based cloth animation has gained much attention from researchers in the last two decades, due to the challenges of realism and performance placed by the film and game industries, as well as by the applications of virtual reality and e-commerce. Consequently, there have been overwhelming achievements in this area, and several approaches have been proposed for modeling and simulating the dynamic behavior of cloth. Modeling cloth as a mass-spring system has been widely used, because of its simplicity and since it leads to more efficient numerical solutions to the governing differential equations. These numerical integration methods are divided into two main categories: explicit integration methods and implicit integration methods. A deeper understanding about their use in cloth simulations is still in order. Thus, the objective of this work is to analyze the integration methods most commonly used in the literature of cloth simulation, based on particle systems, to understand the advantages and disadvantages of their use. A comparative study of those methods is done, considering the criteria of animation realism, computational effort and numerical stability.

Nesta tese consideramos uma nova classe de sistemas markovianos de partículas com infinitas componentes interagentes. O sistema representa a evolução temporal dos potenciais de membrana de um conjunto infinito de neurônios interagentes. Provamos a existência e unicidade do processo construindo um pseudo-algoritmo de simulação perfeita e mostrando que este algoritmo roda em um número finito de passos quase certamente. Estudamos também o comportamento do sistema quando consideramos apenas um conjunto finito de neurônios. Neste caso, construímos um procedimento de simulação perfeita para o acoplamento entre o processo limitado a um conjunto finito de neurônios e o processo que considera todos os neurônios do sistema. Como consequência encontramos um limitante superior para a probabilidade de discrepância entre os processos. / We consider a new class of interacting particle systems with a countable number of interacting components. The system represents the time evolution of the membrane potentials of an infinite set of interacting neurons. We prove the existence and uniqueness of the process, by the construction of a perfect simulation procedure. We show that this algorithm is successful, that is, we show that the number of steps of the algorithm is finite almost surely. We also study the behaviour of the system when we consider only a finite number of neurons. In this case, we construct a perfect simulation procedure for the coupling of the process with a finite number of neurons and the process with a infinite number of neurons. As a consequence we obtain an upper bound for the error we make when sampling from a finite set of neurons instead of the infinite set of neurons.

Context. Particle systems and particle effects are used to simulate a realistic and appealing atmosphere in many virtual environments. However, they do occupy a significant amount of computational resources. The demand for more advanced graphics increases by each generation, likewise does particle systems need to become increasingly more detailed. Objectives. This thesis proposes a texture-based 3D vector field particle system, computed on the Graphics Processing Unit, and compares it to an equation-based particle system. Methods. Several tests were conducted comparing different situations and parameters for the methods. All of the tests measured the computational time needed to execute the different methods. Results. We show that the texture-based method was effective in very specific situations where it was expected to outperform the equation-based. Otherwise, the equation-based particle system is still the most efficient. Conclusions. Generally the equation-based method is preferred, except for in very specific cases. The texture-based is most efficient to use for static particle systems and when a huge number of forces is applied to a particle system. Texture-based vector fields is hardly useful otherwise.

This thesis is split in two different topics. Firstly, we study anomalous transport from kinetic models. Secondly, we consider the equations coming from weak wave turbulence theory and we study them via mean-field limits of finite stochastic particle systems. $\textbf{Anomalous transport from kinetic models.}$ The goal is to understand how fractional diffusion arises from kinetic equations. We explain how fractional diffusion corresponds to anomalous transport and its relation to the classical diffusion equation. In previous works it has been seen that particles systems undergoing free transport and scattering with the media can give rise to fractional phenomena in two cases: firstly, if in the dynamics of the particles there is a heavy-tail equilibrium distribution; and secondly, if the scattering rate is degenerate for small velocities. We use these known results in the literature to study the emergence of fractional phenomena for some particular kinetic equations. Firstly, we study BGK-type equations conserving not only mass (as in previous results), but also momentum and energy. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density making use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momentum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavy-tailed equilibria. Secondly, we will study diffusion phenomena arising from transport of energy in an anharmonic chain. More precisely, we will consider the so-called FPU-$\beta$ chain, which is a very simple model for a one-dimensional crystal in which atoms are coupled to their nearest neighbours by a harmonic potential, weakly perturbed by a nonlinear quartic potential. The starting point of our mathematical analysis is a kinetic equation; lattice vibrations, responsible for heat transport, are modelled by an interacting gas of phonons whose evolution is described by the Boltzmann Phonon Equation. Our main result is the derivation of an anomalous diffusion equation for the temperature. $\textbf{Weak wave turbulence theory and mean-field limits for stochastic particle systems.}$ The isotropic 4-wave kinetic equation is considered in its weak formulation using model homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting. We also consider finite stochastic particle systems undergoing instantaneous coagulation-fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit).

We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.

Dans cette thèse, on étudie le phénomène de localisation d'Anderson des opérateurs de Schrödinger à N particules qui englobe aussi bien la localisation exponentielle des fonctions propres que la localisation dynamique. Dans un premier temps, on considère le modèle d'Anderson discret multi- particulaire avec un potentiel aléatoire à valeurs indépendantes et identiquement distribuées i.i.d. dont la distribution commune est au moins log-Höldérienne et une interaction de courte portée. On établit pour ce modèle la localisation d'Anderson pour les énergies susamment proches du bas du spectre après avoir montré qu'il est non-aléatoire. D'autre part, on montre que la localisation complète des systèmes mono- particulaires s'étend aux systèmes multi-particulaires ayant une interaction globale susamment faible et pour un désordre arbitraire. Pour ce résultat, l'existence d'une densité bornée de la distribution commune des variables aléatoires i.i.d. est nécessaire et on le montre pour des interactions ayant une forme très générale mais bornées. Les résultats sont démontrés à l'aide de l'adaptation aux systèmes multi- particulaires de l'analyse multi-échelle à énergie variable qui permet de traiter des distributions singulières contrairement à la méthode des moments fractionnaires. Les estimées de Wegner intervenant notamment dans l'analyse multi-échelle sont établies pour des cubes vériant une propriété de séparabilité en utilisant le Lemme de Stollmann. On démontre également l'estimée de Combes-Thomas qui joue un rôle important dans l'analyse des énergies extrêmes. / In this thesis, we study for the N-particles Schrodinger operators the Anderson localization phenomenon which consists of both exponential localization of eigenfunctions and dynamical localization. We rst consider the discrete multi-particle Anderson model with a short range interaction and a random potential whose values are independent and identically distributed i.i.d. with a log-Hölder continuous common probability distribution function. For such model, we show that the bottom of its spectrum is non-random and prove the Anderson localization for energies suciently close to the spectral edge. On the other hand, we establish that the complete localization from singleparticle systems extends to multi-particle systems with suciently weak interaction at arbitrary disorder and for absolute continuous probability distribution function of the i.i.d random variables. The results are proved by an adaptation to multi-particle systems of the vari- able energy multi-scale analysis which allows singular distributions instead of the fractional moments method. Wegner bounds, useful for the multi-scale analysis are proved for separable cubes using the Stollmann's Lemma. We also prove the Combes-Thomas estimate which plays an important role in the analysis of extreme energies.