Phase segregated states in a simple three-state stochastic lattice gas are investigated. A two dimensional finite lattice with periodic boundary conditions is filled with one hole and two oppositely "charged" species of particles, subject to an excluded volume constraint. Starting from a completely disordered initial configuration, a sufficiently large external "electric" field <I>E</I> induces the phase segregation, by separating the charges into two strips and "trapping" the hole at an interface between them. Focusing on the steady state, the scaling properties of an appropriate order parameter, depending on drive and system size, are investigated by mean-field theory and Monte Carlo methods. Density profiles of the two interfaces in the ordered system are studied with the help of Monte Carlo simulations and are found to scale in the field-dependent variable, Ε = 2 tanh <I>E</I> /2), for <I>E</I> ≲ 0.8. For larger values of <I>E</I>, independent approximations of the interfacial profiles, obtained within the framework of mean-field theory, exhibit significant deviations from the Monte Carlo data. Interestingly, the deviations can be reduced significantly by a slight modification of the mean-field theory. / Master of Science

Nesta dissertação, estudou-se um modelo tipo gás de rede em três dimensões para simular sistemas líquidos macroscópicos. Aplicou-se o modelo para o estudo das energias e forças envolvidas durante o processo de formação e ruptura de pontes líquidas entre duas placas planas. Esse estudo foi motivado por processos fisiológicos que acontecem no interior dos pulmões dos mamíferos. Além disso, foi feito um estudo das propriedades termodinâmicas do modelo. Com relação a aplicação fisiológica, observou-se que, no processo da formação da ponte líquida, a energia livre da ponte líquida é menor que a energia livre da gota, para diferentes sistemas líquidos. Com este resultado, fez a hipótese de que parte dessa energia é dissipada na forma de energia acústica. A emissão do som também deve ocorrer na ruptura da ponte líquida. Comparando a energia livre no processo de formação e ruptura da ponte líquida observou-se uma curva de histerese. Também foi verificado que para sistemas pequenos, a ponte líquida no modelo computacional se forma antes da previsão analítica. Para a análise termodinâmica, o modelo foi simplificado removendo as placas planas. Foi estudado o caso mais simples desse modelo que continha apenas duas partículas de líquido. Neste caso, calculou-se o calor específico e a energia interna numericamente, e esses resultados foram comparados com cálculos analíticos, validando o modelo numérico. Posteriormente, realizou-se um estudo da transição de fase desse sistema. Em seguida, a energia livre e a força da ponte líquida sobre as placas foram estudadas para diferentes temperaturas utilizando duas metodologias. Na primeira metodologia a entropia foi desprezada, na segunda metodologia, foi utilizando o método ``Overlapping Distribution\'\' que considera a entropia do sistema. Foi concluído que a entropia tem um efeito muito pequeno nas condições estudadas. O modelo é viável para a modelagem de fluidos a nível macroscópico e que portanto pode ser utilizado para quantificar não só as forças internas de estruturas pulmonares como também avaliar as energias liberadas após o processo de ruptura ou formação dessas pontes. / This work studied a three dimension lattice gas model to simulate macroscopic liquid systems. We used the model to study the energy and the forces involved during the process of liquid bridge formation and rupture between two parallel planes. The motivarion of this study was a physiological processes which occur inside the mammals lungs. Furthermore, a study was made to elucidate thermodynamic properties of the model. Concerning to physiological application, it was observed that the free energy of liquid bridge is smaller than the free energy of the droplet, for different liquid systems. With this observation, was proposed that this energy is dissipated as an acoustic energy. This sound should also exist in the rupture of liquid bridge. Comparing the free energy of liquid bridge in the formation and rupture process was observed a hysteresis curve. It was also found an effect of finite size in the formation of small size of the liquid bridge. In the numerical model, the liquid bridge is always formed earlier than expected from the analytical model. In the thermodynamic study, the model was simplified removing both parallel planes. First, the simplest case of this model was studied, only two liquid particles in a large gas lattice. In this case, the specific heat and internal energy was numerically studied and the results was compared with analytical calculation. Subsequently, we carried out a study of the phase transition of this system. Then, the free energy and the force generated between two parallel planes due the presence of the liquid bridge. This studied was performed using two different temperatures and two distinct methods. In the first method the entropy was neglected, and in the second method not. The second method was the ``Overlapping Distribution\'\'. It was concluded that the entropy has a very small effect in the studied conditions. The model is viable for modeling fluids at macroscopic level and therefore can be used to quantify not only the internal forces of the lung structures, but evaluate the energies released after the rupture process of the formation of these bridges.

O fenômeno da capilaridade na escala macroscópica é descrito pela teoria capilar (TC) que se utiliza de superfícies contínuas para modelar as interfaces formadas entre dois meios, sendo um líquido e o outro líquido, gasoso, sólido. A TC é empregada em diversas áreas da biologia, ambientes de microgravidade e em aplicações na escala nanométrica, como no microscópio de força atômica. Essa aproximação por superfícies contínuas pode não ser adequada para sistemas na escala nanométrica, em que são reportados comportamentos anômalos como no preenchimento de líquidos em nanocanais e nanotubos de carbono, oscilações nas medidas de força de adesão capilar e grandes valores de pressões de Laplace negativas. Esses fatos motivam o estudo do fenômeno da capilaridade na escala nanométrica por meio de simulações computacionais. Aqui, utilizamos a dinâmica molecular para estudar a interface de gotas e pontes capilares constituídas de água do modelo SPC/E com volumes da ordem de 100 nanômetros cúbicos e aderidas a placas de cristobalita hidrofóbicas/hidrofílicas. Comparamos as propriedades dessas gotas e pontes capilares com as previsões da TC macroscópica, que são baseadas nos ajustes dos perfis e em cálculos analíticos. Especificamente, confrontamos os perfis das interfaces, os ângulos de contato, as forças de adesão capilar, as pressões de Laplace e o valor da tensão superficial da água. Essas análises foram divididas em três etapas. Na primeira etapa, estudamos as gotas e pontes capilares com simetrias axial e translacional, em que a altura da ponte capilar permaneceu constante. Na segunda etapa, focamos nossos estudos nas pontes capilares com simetria axial (ponte SA) e estudamos o processo de ruptura dessa. Finalmente, na terceira etapa, estudamos as flutuações, que não são previstas pela TC, em sistemas mais simples, como no caso de gotas livres, que não estão aderidas a placas, e em gotas com simetria axial. Mostramos que a TC macroscópica é capaz de explicar satisfatoriamente sistemas com volumes da ordem de 100 nanômetros cúbicos, em que submetemos nossos resultados a comparações rigorosas das soluções analíticas da TC, sendo essa capaz de prever a dependência do ângulo de contato nas alturas das rupturas das ponte SA e os volumes das gotas formadas após a ruptura. / The capillarity phenomenon at macroscopic scale are described by the capillarity theory (CT), which uses continuous surfaces to model the interfaces formed between two media, wherever one medium is liquid and the other can be liquid, gas or solid. The CT is employed in several areas ranging from biology, microgravity environments and applications on the nanoscale, such as in the atomic force microscope. However, the continuous approach may not be adequate for systems at nanoscale, where anomalous behaviors have been reported, such as the filling of liquids in nanochannels and carbon nanotubes, oscillations in measurements of capillary adhesion force and large negative values of Laplace pressures. These facts motivate the study of capillarity phenomenon at the nanometric scale by computational simulations. Here, we use the molecular dynamics to study the droplets and capillary bridges interfaces composed of SPC/E water model and volumes in the order of 100 cubic nanometer, and attached to hydrophobic/hydrophilic cristobalite walls. We have compared the droplets and capillary bridges properties with the macroscopic CT predictions, which are based on profile fitting and analytic calculations. Specifically, we have compared the interface profiles, the contact angles, the capillary adhesion forces, the Laplace pressures and the water surface tension. These analyzes were divided into three steps. In the first step, we have studied droplets and capillary bridges with axial and translational symmetries, where the capillary bridge height remained constant. In the second step, we have focused our studies on capillary bridges with axial symmetry (AS bridge), and we have studied the bridges rupture process. Finally, in the third step, we have studied the fluctuations, which are not predicted by the CT, in simpler systems, such as free droplets, which are not attached to walls, and droplets with axial symmetry. We have shown that the macroscopic CT is able to satisfactorily predict systems with volumes in the order of 100 cubic nanometer, in which we have been submitted our results to rigorous comparisons to the analytic CT solutions, which is able to predict the dependence of contact angle on the AS bridge rupture heights, and the volumes of droplets formed after rupture.

Nous étudions la dynamique d'un traceur soumis à une force extérieure dans un bain de particules. Nous proposons un modèle qui prend en compte explicitement la dynamique du bain, et qui décrit les corrélations entre la dynamique du traceur et la réponse du bain. Nous considérons un traceur biaisé dans un gaz de coeurs durs sur réseau : le traceur réalise une marche aléatoire biaisée tandis que les particules du bain réalisent des marches aléatoires symétriques. Nous étudions plus particulièrement les fluctuations de la position du traceur. Dans la limite de haute densité, nous obtenons des résultats exacts à l'ordre dominant en la densité de lacunes. En géométrie confinée, un calcul analytique des fluctuations de la position du traceur prévoit un long régime superdiffusif, et une transition vers un régime diffusif final. Nous proposons une description simplifiée du système qui révèle le mécanisme physique à l'origine de ce comportement anormal. Nous montrons l'existence d'une anomalie de la vitesse du traceur dans les systèmes quasi-1D. Nous étudions également le cas général d'une densité arbitraire de particules sur un réseau en contact avec un réservoir. Cette situation constitue un problème à N corps décrit par une équation maîtresse, qui ne peut être résolue qu'en recourant à une approximation de type champ moyen consistant en le découplage de certaines fonctions de corrélation. Il est alors possible de déterminer des valeurs approchées de la vitesse, de coefficient de diffusion du traceur ainsi que de la distribution de position du traceur. Nous montrons enfin que l'approximation de découplage est exacte dans les limites de basse et de haute densité. / We study the dynamics of a tracer submitted to an external force in a bath of particles. We propose a model which takes explicitly into account the dynamics of the bath, and which describes the correlations between the dynamics of the tracer and the response of the bath. We consider a biased tracer in a lattice gas of hardcore particles: the tracer performs a biased random walk whereas the bath particles perform symmetric random walks. We study in particular the fluctuations of the position of the tracer. In the high-density limit, we obtain exact results at leading order in the density of vacancies. In confined geometries, an analytical calculation of the fluctuations of tracer position predicts a long superdiffusive regime, and a crossover to an ultimate diffusive regime. We give a simplified description of the system that unveils the physical mechanism explaining this anomalous behavior. We show the existence of a velocity anomaly in quasi-1D systems.We also study the general case of an arbitrary density of particles on a lattice in contact with a reservoir. This situation is a N-body problem described by a master equation, that can be solved by resorting to a mean-field-type approximation, which consists in the decoupling of relevant correlation functions. It is then possible to determine approximate values of the velocity, the diffusion coefficient and the distribution of the position of the tracer. We finally show that the decoupling approximation is exact in the high-density and low-density limits.

O fenômeno da capilaridade na escala macroscópica é descrito pela teoria capilar (TC) que se utiliza de superfícies contínuas para modelar as interfaces formadas entre dois meios, sendo um líquido e o outro líquido, gasoso, sólido. A TC é empregada em diversas áreas da biologia, ambientes de microgravidade e em aplicações na escala nanométrica, como no microscópio de força atômica. Essa aproximação por superfícies contínuas pode não ser adequada para sistemas na escala nanométrica, em que são reportados comportamentos anômalos como no preenchimento de líquidos em nanocanais e nanotubos de carbono, oscilações nas medidas de força de adesão capilar e grandes valores de pressões de Laplace negativas. Esses fatos motivam o estudo do fenômeno da capilaridade na escala nanométrica por meio de simulações computacionais. Aqui, utilizamos a dinâmica molecular para estudar a interface de gotas e pontes capilares constituídas de água do modelo SPC/E com volumes da ordem de 100 nanômetros cúbicos e aderidas a placas de cristobalita hidrofóbicas/hidrofílicas. Comparamos as propriedades dessas gotas e pontes capilares com as previsões da TC macroscópica, que são baseadas nos ajustes dos perfis e em cálculos analíticos. Especificamente, confrontamos os perfis das interfaces, os ângulos de contato, as forças de adesão capilar, as pressões de Laplace e o valor da tensão superficial da água. Essas análises foram divididas em três etapas. Na primeira etapa, estudamos as gotas e pontes capilares com simetrias axial e translacional, em que a altura da ponte capilar permaneceu constante. Na segunda etapa, focamos nossos estudos nas pontes capilares com simetria axial (ponte SA) e estudamos o processo de ruptura dessa. Finalmente, na terceira etapa, estudamos as flutuações, que não são previstas pela TC, em sistemas mais simples, como no caso de gotas livres, que não estão aderidas a placas, e em gotas com simetria axial. Mostramos que a TC macroscópica é capaz de explicar satisfatoriamente sistemas com volumes da ordem de 100 nanômetros cúbicos, em que submetemos nossos resultados a comparações rigorosas das soluções analíticas da TC, sendo essa capaz de prever a dependência do ângulo de contato nas alturas das rupturas das ponte SA e os volumes das gotas formadas após a ruptura. / The capillarity phenomenon at macroscopic scale are described by the capillarity theory (CT), which uses continuous surfaces to model the interfaces formed between two media, wherever one medium is liquid and the other can be liquid, gas or solid. The CT is employed in several areas ranging from biology, microgravity environments and applications on the nanoscale, such as in the atomic force microscope. However, the continuous approach may not be adequate for systems at nanoscale, where anomalous behaviors have been reported, such as the filling of liquids in nanochannels and carbon nanotubes, oscillations in measurements of capillary adhesion force and large negative values of Laplace pressures. These facts motivate the study of capillarity phenomenon at the nanometric scale by computational simulations. Here, we use the molecular dynamics to study the droplets and capillary bridges interfaces composed of SPC/E water model and volumes in the order of 100 cubic nanometer, and attached to hydrophobic/hydrophilic cristobalite walls. We have compared the droplets and capillary bridges properties with the macroscopic CT predictions, which are based on profile fitting and analytic calculations. Specifically, we have compared the interface profiles, the contact angles, the capillary adhesion forces, the Laplace pressures and the water surface tension. These analyzes were divided into three steps. In the first step, we have studied droplets and capillary bridges with axial and translational symmetries, where the capillary bridge height remained constant. In the second step, we have focused our studies on capillary bridges with axial symmetry (AS bridge), and we have studied the bridges rupture process. Finally, in the third step, we have studied the fluctuations, which are not predicted by the CT, in simpler systems, such as free droplets, which are not attached to walls, and droplets with axial symmetry. We have shown that the macroscopic CT is able to satisfactorily predict systems with volumes in the order of 100 cubic nanometer, in which we have been submitted our results to rigorous comparisons to the analytic CT solutions, which is able to predict the dependence of contact angle on the AS bridge rupture heights, and the volumes of droplets formed after rupture.

Nesta dissertação, estudou-se um modelo tipo gás de rede em três dimensões para simular sistemas líquidos macroscópicos. Aplicou-se o modelo para o estudo das energias e forças envolvidas durante o processo de formação e ruptura de pontes líquidas entre duas placas planas. Esse estudo foi motivado por processos fisiológicos que acontecem no interior dos pulmões dos mamíferos. Além disso, foi feito um estudo das propriedades termodinâmicas do modelo. Com relação a aplicação fisiológica, observou-se que, no processo da formação da ponte líquida, a energia livre da ponte líquida é menor que a energia livre da gota, para diferentes sistemas líquidos. Com este resultado, fez a hipótese de que parte dessa energia é dissipada na forma de energia acústica. A emissão do som também deve ocorrer na ruptura da ponte líquida. Comparando a energia livre no processo de formação e ruptura da ponte líquida observou-se uma curva de histerese. Também foi verificado que para sistemas pequenos, a ponte líquida no modelo computacional se forma antes da previsão analítica. Para a análise termodinâmica, o modelo foi simplificado removendo as placas planas. Foi estudado o caso mais simples desse modelo que continha apenas duas partículas de líquido. Neste caso, calculou-se o calor específico e a energia interna numericamente, e esses resultados foram comparados com cálculos analíticos, validando o modelo numérico. Posteriormente, realizou-se um estudo da transição de fase desse sistema. Em seguida, a energia livre e a força da ponte líquida sobre as placas foram estudadas para diferentes temperaturas utilizando duas metodologias. Na primeira metodologia a entropia foi desprezada, na segunda metodologia, foi utilizando o método ``Overlapping Distribution\'\' que considera a entropia do sistema. Foi concluído que a entropia tem um efeito muito pequeno nas condições estudadas. O modelo é viável para a modelagem de fluidos a nível macroscópico e que portanto pode ser utilizado para quantificar não só as forças internas de estruturas pulmonares como também avaliar as energias liberadas após o processo de ruptura ou formação dessas pontes. / This work studied a three dimension lattice gas model to simulate macroscopic liquid systems. We used the model to study the energy and the forces involved during the process of liquid bridge formation and rupture between two parallel planes. The motivarion of this study was a physiological processes which occur inside the mammals lungs. Furthermore, a study was made to elucidate thermodynamic properties of the model. Concerning to physiological application, it was observed that the free energy of liquid bridge is smaller than the free energy of the droplet, for different liquid systems. With this observation, was proposed that this energy is dissipated as an acoustic energy. This sound should also exist in the rupture of liquid bridge. Comparing the free energy of liquid bridge in the formation and rupture process was observed a hysteresis curve. It was also found an effect of finite size in the formation of small size of the liquid bridge. In the numerical model, the liquid bridge is always formed earlier than expected from the analytical model. In the thermodynamic study, the model was simplified removing both parallel planes. First, the simplest case of this model was studied, only two liquid particles in a large gas lattice. In this case, the specific heat and internal energy was numerically studied and the results was compared with analytical calculation. Subsequently, we carried out a study of the phase transition of this system. Then, the free energy and the force generated between two parallel planes due the presence of the liquid bridge. This studied was performed using two different temperatures and two distinct methods. In the first method the entropy was neglected, and in the second method not. The second method was the ``Overlapping Distribution\'\'. It was concluded that the entropy has a very small effect in the studied conditions. The model is viable for modeling fluids at macroscopic level and therefore can be used to quantify not only the internal forces of the lung structures, but evaluate the energies released after the rupture process of the formation of these bridges.

Dans cette thèse, nous étudions la robustesse dans le contexte de la modélisation de systèmes complexes par les automates cellulaires. En effet, si l'on cherche à reproduire un comportement émergent à partir d'un modèle d'automate cellulaire, il nous semble nécessaire de se demander si les comportements observés sont bien le résultat d'interactions entre entités constituantes, ou bien s'ils dépendent d'une définition particulière du modèle. Nous allons ainsi être amenés à considérer la robustesse du modèle, à savoir la résistance de son comportement à de petites variations sur les attributs de sa définition. Dans un premier temps, nous montrons la pertinence de cette approche en considérant plusieurs définitions possibles d'une perturbation de la mise à jour globale et en les appliquant à une classe simple et représentative de modèles d'automates cellulaires, les Automates Cellulaires Elémentaires. Nous observons que, malgré le fait que nos perturbations soient proches et qu'une majorité des modèles considérés ne change pas de comportement, quelques cas particuliers montrent des changements qualitatifs du comportement que nous étudions plus en détail. Dans un second temps, nous appliquons cette approche en nous penchant sur un modèle particulier d'automate cellulaire, qui simule le phénomène de formation d'essaim à partir d'un modèle évolué d'automate cellulaire, le gaz sur réseau. Nous explorons la robustesse du comportement du modèle en considérant la perturbation de deux attributs du modèle, la forme de la grille cellulaire et la mise à jour globale, et en tirons les conclusions sur la relation entre l'observation du comportement et la définition précise du modèle / In this thesis, we study the role of robustness in the context of the modelling of complex systems by cellular automata. Indeed, if we consider a cellular automaton which aims at reproducing an emergent behaviour from a similar structure, we want to determine whether its observed dynamics are the result of the interaction of entities, or whether it depends a precise definition of the model. We thus consider the model's robustness, that is, the resistance of the behaviour to small perturbations on the model features. First, we show the relevance of this approach by considering several definitions of a perturbation of the global updating and by applying them to a simple and representative class of cellular automata, the Elementary Cellular Automata. We observe that, despite the fact that most models show little or no change between the different perturbations, some particular cases show qualitative changes that we study in detail. Second, we apply this approach to a particular model of cellular automata, which simulates a swarming behaviour based on a lattice-gas model. We then explore the model robustness by considering the pertubations of two of the model's attributes, the lattice shape and the global updating, and discuss the relationship between the observation of the behaviour and the precise definitions of the model

Orientador: Siome Klein Goldenstein / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-16T21:25:11Z (GMT). No. of bitstreams: 1 Saboia_PriscilaCorrea_M.pdf: 2582894 bytes, checksum: 1645aa6b8b779ee7180adbbb04d23981 (MD5) Previous issue date: 2010 / Resumo: Atualmente, estima-se que a população mundial seja de cerca de seis bilhões e oitocentos milhões de habitantes (6.800.000.000), dos quais metade mora em aglomerados urbanos. Nestes centros, é corriqueiro o fen¿omeno da movimentação de multidões. Tecnicamente, entende-se multidão como um grande grupo de indivíduos em um mesmo ambiente físico, compartilhando um objetivo comum e podendo agir diferentemente do que quando estão sozinhos. Compreender a movimentação destas multidões é de vital import¿ancia para o planejamento e a melhoria dos locais públicos, não só no sentido de facilitar e agilizar o deslocamento dos cidadãos, mas também garantir-lhes segurança, especialmente em condições de perigo iminente, onde pode haver a necessidade de evacuação de tais locais. Além disso, existem várias áreas do conhecimento que têm potencial para se beneficiar do estudo do comportamento de multidões. Na indústria de entretenimento, por exemplo, simulações de multidões podem ser utilizadas na produção de animações e jogos de computador. No treinamento policial e militar, simulações podem ser usadas para demonstração e controle de rebeliões. Na área de engenharia de segurança, simulações podem ser utilizadas para estudo de desocupação emergencial de construções, navios e aviões. Em todas as áreas citadas, observa-se que a necessidade por simulações de multidões advém de duas situações que podem ocorrer no mundo real. Primeiramente, pode ser perigoso para os indivíduos realizar as ações objetivadas (como cair de um prédio em um filme, ou evacuar uma sala de cinema em chamas, por exemplo), bem como é antiético submetê-los a tais condições. Segundo, é muito complexo e oneroso lidar com um grande número de indivíduos no mundo real. Ambas as situações podem ser evitadas pela simulação computacional da situação real. Nestes termos, o objetivo desta dissertação é modelar a movimentação de multidões, tendo em vista a simulação em computador. Para tanto, sistemas multiagentes brownianos são introduzidos como uma alternativa tecnológica 'a implementação dos modelos encontrados na literatura, bem como 'a implementação de um novo modelo de movimentação de multidões, híbrido por reunir conceitos de modelos que lançam mão das chamadas forças sociais, com conceitos de modelos baseados na estratégia Lattice-Gas. Como resultado prático, um novo simulador de sistemas multiagentes construído para a tarefa de simulação de movimentação de multidões é apresentado / Abstract: Currently, it is estimated that world population is about six billion and eight hundred million inhabitants (6.8 billion), of whom half live in urban areas. In these centers, it is common the phenomenon of moving crowds. Technically, a crowd can be seen as a large group of individuals put in the same physical environment, sharing a common goal and acting differently than when they are alone. Understanding the movement of these crowds is very important for planning and improving public places, not only in order to facilitate and expedite the movement of citizens, but also in order to guarantee their safety, especially in conditions of imminent danger, where it can be necessary the evacuation of such sites. Furthermore, there are several areas of knowledge that can gain benefits from the study of crowd behavior. In the entertainment industry, for example, crowd simulations can be used to produce animations and computer games. In Police and military training, simulations can be used for demonstration and control of riots. In the area of safety engineering, simulations can be used to study the urgent evacuation of buildings, ships and aircrafts. In all mentioned areas, it is observed that the need for crowd simulations comes from two situations, taking into consideration the real world. First, it can be dangerous for individuals to perform the desired actions (like falling from a building in a movie, or evacuating the room on fire of a failing movie session). It is also unethical to expose them to such conditions. Second, it is very complex and expensive to handle a large number of individuals in the real world. Both situations can be dealt by simulating the real world into a computer. Thus, this dissertation aims to model the movement of crowds, always having in mind the computer simulation. For this, Brownian multi-agent systems are introduced as a technological alternative to implement the models found in literature, as well as to implement a new hybrid movement model, that gathers together concepts from solutions based on social forces, and solutions based on Lattice- Gas. As a practical result, a new simulator for multi-agent systems is presented, built for the task of simulating moving crowds / Mestrado / Sistemas de Informação / Mestre em Ciência da Computação

In this work we provide several improvements in the study of phase transitions of interacting particle systems: - We determine a quantitative relation between non-extremality of the limiting Gibbs measure of a tree-based spin system, and the temporal mixing of the Glauber Dynamics over its finite projections. We define the concept of 'sensitivity' of a reconstruction scheme to establish such a relation. In particular, we focus on the independent sets model, determining a phase transition for the mixing time of the Glauber dynamics at the same location of the extremality threshold of the simple invariant Gibbs version of the model. - We develop the technical analysis of the so-called spatial mixing conditions for interacting particle systems to account for the connectivity structure of the underlying graph. This analysis leads to improvements regarding the location of the uniqueness/non-uniqueness phase transition for the independent sets model over amenable graphs; among them, the elusive hard-square model in lattice statistics, which has received attention since Baxter's solution of the analogous hard-hexagon in 1980. - We build on the work of Montanari and Gerschenfeld to determine the existence of correlations for the coloring model in sparse random graphs. In particular, we prove that correlations exist above the 'clustering' threshold of such a model; thus providing further evidence for the conjectural algorithmic 'hardness' occurring at such a point.

Cancer cells display characteristic traits acquired in a step-wise manner during carcinogenesis. Some of these traits are autonomous growth, induction of angiogenesis, invasion and metastasis. In this thesis, the focus is on one of the latest stages of tumor progression, tumor invasion. Tumor invasion emerges from the combined effect of tumor cell-cell and cell-microenvironment interactions, which can be studied with the help of mathematical analysis. Cellular automata (CA) can be viewed as simple models of self-organizing complex systems in which collective behavior can emerge out of an ensemble of many interacting &amp;quot;simple&amp;quot; components. In particular, we focus on an important class of CA, the so-called lattice-gas cellular automata (LGCA). In contrast to traditional CA, LGCA provide a straightforward and intuitive implementation of particle transport and interactions. Additionally, the structure of LGCA facilitates the mathematical analysis of their behavior. Here, the principal tools of mathematical analysis of LGCA are the mean-field approximation and the corresponding Lattice Boltzmann equation. The main objective of this thesis is to investigate important aspects of tumor invasion, under the microscope of mathematical modeling and analysis: Impact of the tumor environment: We introduce a LGCA as a microscopic model of tumor cell migration together with a mathematical description of different tumor environments. We study the impact of the various tumor environments (such as extracellular matrix) on tumor cell migration by estimating the tumor cell dispersion speed for a given environment. Effect of tumor cell proliferation and migration: We study the effect of tumor cell proliferation and migration on the tumor’s invasive behavior by developing a simplified LGCA model of tumor growth. In particular, we derive the corresponding macroscopic dynamics and we calculate the tumor’s invasion speed in terms of tumor cell proliferation and migration rates. Moreover, we calculate the width of the invasive zone, where the majority of mitotic activity is concentrated, and it is found to be proportional to the invasion speed. Mechanisms of tumor invasion emergence: We investigate the mechanisms for the emergence of tumor invasion in the course of cancer progression. We conclude that the response of a microscopic intracellular mechanism (migration/proliferation dichotomy) to oxygen shortage, i.e. hypoxia, maybe responsible for the transition from a benign (proliferative) to a malignant (invasive) tumor. Computing in vivo tumor invasion: Finally, we propose an evolutionary algorithm that estimates the parameters of a tumor growth LGCA model based on time-series of patient medical data (in particular Magnetic Resonance and Diffusion Tensor Imaging data). These parameters may allow to reproduce clinically relevant tumor growth scenarios for a specific patient, providing a prediction of the tumor growth at a later time stage. / Krebszellen zeigen charakteristische Merkmale, die sie in einem schrittweisen Vorgang während der Karzinogenese erworben haben. Einige dieser Merkmale sind autonomes Wachstum, die Induktion von Angiogenese, Invasion und Metastasis. Der Schwerpunkt dieser Arbeit liegt auf der Tumorinvasion, einer der letzten Phasen der Tumorprogression. Die Tumorinvasion ensteht aus der kombinierten Wirkung von den Wechselwirkungen Tumorzelle-Zelle und Zelle-Mikroumgebung, die mit die Hilfe von mathematischer Analyse untersucht werden können. Zelluläre Automaten (CA) können als einfache Modelle von selbst-organisierenden komplexen Systemen betrachtet werden, in denen kollektives Verhalten aus einer Kombination von vielen interagierenden &amp;quot;einfachen&amp;quot; Komponenten entstehen kann. Insbesondere konzentrieren wir uns auf eine wichtige CA-Klasse, die sogenannten Zelluläre Gitter-Gas Automaten (LGCA). Im Gegensatz zu traditionellen CA bieten LGCA eine einfache und intuitive Umsetzung der Teilchen und Wechselwirkungen. Zusätzlich erleichtert die Struktur der LGCA die mathematische Analyse ihres Verhaltens. Die wichtigsten Werkzeuge der mathematischen Analyse der LGCA sind hier die Mean-field Approximation und die entsprechende Lattice - Boltzmann - Gleichung. Das wichtigste Ziel dieser Arbeit ist es, wichtige Aspekte der Tumorinvasion unter dem Mikroskop der mathematischen Modellierung und Analyse zu erforschen: Auswirkungen der Tumorumgebung: Wir stellen einen LGCA als mikroskopisches Modell der Tumorzellen-Migration in Verbindung mit einer mathematischen Beschreibung der verschiedenen Tumorumgebungen vor. Wir untersuchen die Auswirkungen der verschiedenen Tumorumgebungen (z. B. extrazellulären Matrix) auf die Migration von Tumorzellen dürch Schätzung der Tumorzellen-Dispersionsgeschwindigkeit in einem gegebenen Umfeld. Wirkung von Tumor-Zellenproliferation und Migration: Wir untersuchen die Wirkung von Tumorzellenproliferation und Migration auf das invasive Verhalten der Tumorzellen durch die Entwicklung eines vereinfachten LGCA Tumorwachstumsmodells. Wir leiten die entsprechende makroskopische Dynamik und berechnen die Tumorinvasionsgeschwindigkeit im Hinblick auf die Tumorzellenproliferation- und Migrationswerte. Darüber hinaus berechnen wir die Breite der invasiven Zone, wo die Mehrheit der mitotischer Aktivität konzentriert ist, und es wird festgestellt, dass diese proportional zu den Invasionsgeschwindigkeit ist. Mechanismen der Tumorinvasion Entstehung: Wir untersuchen Mechanismen, die für die Entstehung von Tumorinvasion im Verlauf des Krebs zuständig sind. Wir kommen zu dem Schluss, dass die Reaktion eines mikroskopischen intrazellulären Mechanismus (Migration/Proliferation Dichotomie) zu Sauerstoffmangel, d.h. Hypoxie, möglicheweise für den Übergang von einem gutartigen (proliferative) zu einer bösartigen (invasive) Tumor verantwortlich ist. Berechnung der in-vivo Tumorinvasion: Schließlich schlagen wir einen evolutionären Algorithmus vor, der die Parameter eines LGCA Modells von Tumorwachstum auf der Grundlage von medizinischen Daten des Patienten für mehrere Zeitpunkte (insbesondere die Magnet-Resonanz-und Diffusion Tensor Imaging Daten) ermöglicht. Diese Parameter erlauben Szenarien für einen klinisch relevanten Tumorwachstum für einen bestimmten Patienten zu reproduzieren, die eine Vorhersage des Tumorwachstums zu einem späteren Zeitpunkt möglich machen.