No presente trabalho provamos resultados sobre as flutuações dos fluxos de partículas e das partículas marcadas no processo de Hammersley multiclasse. Os métodos das demonstrações são robustos, formulados de modo a serem aplicados em outros processos, em particular se aplicam ao processo de exclusão totalmente assimétrico multiclasse (TASEP multiclasse) e à seu respectivo modelo de percolação de última passagem. Os principais teoremas obtidos são um teorema central do limite para o choque, seu coeficiente de difusão e uma fórmula exata para a variância do fluxo de partículas de classe N >1 para o processo em equilíbrio multiclasse. / We prove fluctuations results concerning fluxes of particles and tagged particles on multiclass Hammersley process. The methods used are robust and apply to other processes, in particular all the proofs can be adapted to the Multiclass totally asymmetric simple exclusion process (Multiclass TASEP) and its respective last passage percolation model. The main theorems obtained are a central limit theorem for the shock, its diffusion coefficient and an exact formula for the variance of the $N$-th class particle flux in a stationary version of the multiclass process when N > 1.

Collagen scaffolds are porous structures which are used in bioreactors and in a wide range of tissue engineering applications. In these contexts, the scaffolds may be subjected to conditions in which fluid is forced through the structure and the scaffold is simultaneously compressed. It is clear that fluid transport within collagen scaffolds, and the inter-relationships between permeability, scaffold structure, fluid pressure and scaffold deformation are of key importance. However, these relationships remain poorly understood. In this thesis, a series of isotropic collagen structures were produced using a freeze-drying technique from aqueous slurry concentrations 0.5, 0.75 and 1 wt%, and fully characterised using X-ray micro-tomography and compression testing. It was found that collagen wt% influenced structural parameters such as pore size, porosity, relative density and mechanical properties. Percolation theory was used to investigate the pore interconnectivity of each scaffold. Structures with lower collagen fraction resulted in larger percolation diameters, but lower mechanical stiffness. Aligned collagen scaffolds were also produced by altering the freeze-drying protocol and using different types of mould materials and designs. It was found that a polycarbonate mould with stainless base resulted in vertically aligned structures with low angular variation. When compared with isotropic scaffolds from slurry of the same concentration, aligned scaffolds had a larger percolation diameter. Tortuosity was used as a mathematical tool to characterise the interconnected pathways within each porous structure. The effect of the size of the region of interest (ROI) chosen and the size of the virtual probe particle used in the analysis on the values of tortuosity calculated were determined and an optimised calculation methodology developed. Increasing the collagen fraction within isotropic scaffolds increased the tortuosity, and aligned structures had smaller tortuosity values than their isotropic counterparts. Permeability studies were conducted using two complementary experimental rigs designed to cover a range of pressure regimes and the results were compared with predictions from mathematical models and computational simulations. At low pressures, it was found that the lower collagen fraction structures, which had more open morphologies, had higher permeabilities. Alignment of the structure also enhanced permeability. The scaffolds all experienced deformation at high pressures resulting in a restriction of fluid flow. The lower collagen fraction scaffolds experienced a sharper decrease in permeability with increased pressure and aligned structures were more responsive to deformation than their isotropic counterparts. The inter-relationships between permeability, scaffold structure, fluid pressure and deformation of collagen scaffolds were explored. For isotropic samples, permeability followed a broad $(1- \epsilon)^2$ behaviour with strain as predicted by a tetrakaidecahedral structural model, with the constant of proportionality changing with collagen fraction. In contrast, the aligned structures did not follow this behaviour with the permeability dropping much more sharply in the early stages of compression. Open-cell polyurethane (PU) foams, sometimes used as dressings in wound healing applications, are often compared with collagen scaffolds in permeability models and were used in this thesis as a comparison structure. The foam had a higher permeability than the scaffolds due to its larger pore sizes and higher interconnectivity. In the light of the effects of compression on permeability, the changes in porous structure with compression were explored in isotropic and aligned 0.75 wt% scaffolds. Unlike the fluid flow experiments, these experiments were carried out in the dry state. Deformation in simple linear compression and in step-wise compression was studied, and the stress relaxation behaviour of the scaffolds characterised. A methodology was developed to characterise the structural changes accompanying compression using X-ray micro-tomography with an in situ compression stage. The methodology accounted for the need for samples to remain unchanged during the scan collection period for stable image reconstruction. The scaffolds were studied in uniaxial compression and biaxial compression and it was found that pore size and percolation diameter decreased with increasing compressive strain, while the tortuosity increased. The aligned structure was less affected than the isotropic at low compressions, in contrast to the results from the permeability study in which the aligned structure was more responsive to strain. This suggests that the degree of hydration may affect the structural changes observed. The insights gained in this study of the inter-relationships between microstructure, fluid dynamics and deformation in collagen scaffolds are of relevance to the informed design of porous structures for medical applications.

Este trabalho visa a estudar o modelo de percolação independente, de Bernoulli, em grafos, tendo como objetivo principal obter condições que garantam a ocorrência de transição de fase. Iniciamos apresentando as definições e algumas técnicas fundamentais para o modelo de percolação (de elos ou de sítios) em um grafo infinito, conectado e localmente finito. Demonstramos então dois resultados essenciais: os fatos do parâmetro crítico não depender da escolha do vértice e da existência de um aglomerado infinito ter probabilidade 0 ou 1. Também obtemos um limitante inferior para o parâmetro crítico quando o grafo é de grau limitado. Para finalizar esta parte introdutória, analisamos a percolação em grafos particulares, a saber, a rede hipercúbica Z^d (para a qual mostramos a existência de transição de fase em dimensão d >= 2 e a unicidade do aglomerado infinito na fase supercrítica) e alguns tipos de árvores (para as quais apresentamos os parâmetros críticos). Na parte mais importante da dissertação, tendo como base os trabalhos de Benjamini e Schramm, de Häggström, Schonmann e Steif e de Lyons e Peres, introduzimos os conceitos de transitividade, amenabilidade e amenabilidade forte para um grafo. Fazemos uma detalhada discussão destas definições: provamos que a constante de Cheeger ancorada não depende do vértice em que é ancorada, estudamos relações entre os conceitos (amenabilidade e amenabilidade forte são noções distintas, bem como condições necessárias e suficientes para ambas) e calculamos a constante de Cheeger e a constante de Cheeger ancorada para alguns grafos. Finalmente, utilizando a técnica de crescimento do aglomerado, apresentamos para a probabilidade crítica um limitante superior que depende da constante ancorada. Isto nos permite concluir que ocorre transição de fase para qualquer grafo infinito, conectado, fracamente não-amenável (de constante de Cheeger ancorada positiva) e de grau limitado. / This work intends to study independent Bernoulli percolation model on graphs; the main purpose is obtaining conditions for phase transition. We begin presenting the definitions and some basic techniques for bond percolation and site percolation models on infinite, connected, locally finite graphs. We prove two essential results: the critical parameter is independent of the choice of the vertex and the probability that there exists an infinite cluster takes the values 0 and 1 only. We also obtain a lower bound for critical parameter when the graph is of bounded degree. To finish this preliminary part, we analyze percolation on particular graphs, namely the d-dimensional cubic lattice Z^d (for which we prove that there exists phase transition in dimension d >= 2 and the uniqueness of the infinite cluster in supercritical phase) and some trees (for which we present the critical parameters). In the most important part of this essay, founded in the works of Benjamini and Schramm, Häggström, Schonmann and Steif and Lyons and Peres, we introduce the concepts of transitivity, amenability and strong amenability. We discuss in detail these definitions: we prove that anchored Cheeger constant does not depend on the choice of the vertex, we study some relations (amenability and strong amenability are distinct notions, and necessary and sufficient conditions for both) and we obtain Cheeger constant and anchored Cheeger constant for some graphs. Finally, using the growing cluster technique, we present for the critical probability an upper bound that depends on the anchored constant. This permits us to conclude that there exists phase transition on infinite, connected, weakly non-amenable graphs of bounded degree.

Estudamos o modelo dos sapos na árvore homogênea, um sistema de partículas a tempo discreto cuja dinâmica é sintetizada a seguir. No instante inicial, existe em cada vértice da árvore um número aleatório independente e identicamente distribuído de partículas; aquelas posicionadas em um vértice fixado estão ativas, as demais inativas. Partículas ativas realizam passeios aleatórios simples, independentes, a tempo discreto, com probabilidade de desaparecimento (1 - p) em cada instante. Uma partícula inativa torna-se ativa assim que seu vértice é visitado por uma partícula ativa. Consideramos nesta tese o valor crítico p_c que separa a fase em que o processo se extingue quase certamente da fase em que existem partículas ativas em todos os instantes com probabilidade positiva. Provamos um limitante superior para a probabilidade crítica p_c, o qual melhora o resultado anteriormente conhecido para o caso de configuração inicial de uma partícula por vértice. O argumento utilizado consiste na descrição do modelo dos sapos como um modelo de percolação orientada que domina processos de ramificação convenientemente definidos. Obtemos também o valor assintótico do limitante superior estabelecido, mostrando ser igual ao valor assintótico da probabilidade crítica. / We study the frog model on the homogeneous tree, a discrete-time particle system whose dynamics is summarized next. Initially there is an independent and identically distributed random number of particles at each vertex of the tree; those placed at a fixed vertex are active, the others being inactive. Active particles perform independent discrete-time simple random walks, with probability of disappearance (1 - p) at each instant. An inactive particle becomes active once its vertex is hit by an active particle. We consider in this thesis the critical value p_c that separates the phase in which the process dies out almost surely from the phase in which there exist active particles at all times with positive probability. We prove an upper bound for the critical probability p_c, which improves the formerly known result for the case of one particle per vertex initial configuration. The employed argument builds on the description of the frog model as an oriented percolation model which dominates suitably defined branching processes. We also obtain the asymptotic value of the stated upper bound, showing that it equals the asymptotic value of the critical probability.

Vários estudos têm mostrado os benefícios da aplicação agronômica de lodo de esgoto sobre as propriedades físicas e químicas do solo. No entanto, poucos são aqueles que avaliam o impacto desta prática sobre o comportamento de pesticidas em solos tropicais. O principal objetivo deste estudo foi avaliar o efeito da adição de diferentes fontes de lodo (Ribeirão Preto, Franca e Jundiaí) na sorção / dessorção e lixiviação do herbicida ametrina em solos canavieiros do estado de São Paulo (Neossolo Quartzarênico Órtico Típico (RQ), Latossolo Vermelho Distrófico Típico (LVd), Argissolo Vermelho Eutroférrico Chernossólico (PV) e Latossolo Vermelho Distroférrico (LVdf). Cinco concentrações (de 4,4 a 79,8 mg i.a. L-1) do herbicida foram utilizadas para obter os valores dos coeficientes de sorção de Freündlich (Kf) e dos coeficientes aparentes de sorção médio e para a menor concentração (Kd ap médio e Kd ap [menor]), a qual corresponde à dose de campo recomendada. Para o teste de lixiviação, o método utilizado foi o de lixiviação em colunas de solo (diâmetro = 5 cm e comprimento = 30 cm), utilizando-se três repetições para cada tratamento, sobre as quais foi simulada chuva de 200 mm uniformemente distribuídas durante 48 h, após a aplicação da solução de 14Cametrina na dose de 3,0 kg i.a. ha-1. De forma geral, a sorção da ametrina variou de moderada à alta em todos os tratamentos (2,68 < Kd ap [menor] < 85,71 L kg-1). Solos argilosos com maior teor de matéria orgânica e argilas do tipo 2:1, como é o caso do PV, apresentaram muito maior potencial de sorção da ametrina; enquanto que solos arenosos com baixos teores de matéria orgânica, como foram os casos do LVd e RQ, apresentaram moderado potencial de sorção. A aplicação de lodos menos estabilizados, com biomassa e material orgânico menos recalcitrante e, portanto, com maiores valores de carbono orgânico total, carbono orgânico dissolvido e pH, como é o caso do lodo de Ribeirão Preto, tendeu a diminuir o potencial de sorção da ametrina. Já a adição de lodos mais recalcitrantes, como o de Jundiaí, tendeu a aumentar o potencial de sorção da ametrina, principalmente em solos arenosos devido a sua menor capacidade tampão. Nestes cenários, poderá ocorrer redução da eficácia agronômica ametrina, uma vez que haverá menos produto disponível na solução do solo. A ametrina apresentou baixo potencial de lixiviação (< 1% da quantidade aplicada) em todos os tratamentos, os quais não apresentaram diferenças entre si, sendo que a grande maioria do pesticida (> 95% da quantidade aplicada) ficou retida na camada de 0-10 cm de profundidade da coluna de solo. Isto implica dizer que a ametrina apresenta baixo potencial de contamina águas subterrâneas, mesmo em solos arenosos, como RQ (90% de areia) / Several studies have shown the benefits of applying sewage sludge on the physical and chemical properties of the soils. However, just a few of them evaluates the impact of this practice on the behavior of pesticides in tropical soils. The main goal of this research was to evaluate the effects of applying different sources of sewage sludge (Ribeirão Preto, Franca e Jundiaí) on the sorption / desorption and leaching of ametryne in soils from São Paulo state (Brazil) cultivated with sugarcane (Neossolo Quartzarênico Órtico Típico (RQ), Latossolo Vermelho Distrófico Típico (LVd), Argissolo Vermelho Eutroférrico Chernossólico (PV) e Latossolo Vermelho Distroférrico (LVdf). Five concentrations (4.4 to 79.8 mg a.i. ha-1) of the herbicide were applied to the soil samples to attain the Freündlich sorption coefficients (Kf) and the apparent sorption coefficient for the lower concentration (Kd ap [menor]), which corresponded to the field application rate. For the leaching test the adopted method was the soil leaching columns (diameter = 5 cm and depth = 30 cm), in triplicates, over which a 200 mm rainfall evenly distributed during 48 h was simulated just after ametryne application at the rate of 3.0 kg a.i. ha-1. In general, ametryne sorption ranged from moderate to high in all treatments (2.68 < Kd ap [menor] < 85.71 L kg-1). Clay soils with higher organic matter and 2:1 clay contents, such as the PV, showed much higher sorption potential, whereas sand soils with low organic matter content, such as LVd and RQ, showed moderate sorption potential. The application of less stabilized sewage sludges, with less recalcitrant biomass and organic material and, therefore, with higher organic matter and dissolved organic carbon contents and pH values, such as the Ribeirão Preto, tends to decrease ametryne sorption potential. Otherwise, the addition of more recalcitrant sludges, such as the Jundiaí, tends to enhance its sorption potential, mainly in sand soils due to its lower buffer capacity. In those cases, the agronomic efficacy of ametryne may be reduced since there is lower concentration of the compound available in the soil solution. The ametryne presented low leaching potential in all treatments (< 1% of the applied amount), which were not different among themselves, and that its majority (> 95% of the applied amount) was found at 0 -10 cm soil depth in the column. It implies that ametryne has low potential to contaminate groundwater even in sandy soils, such as the RQ (90% sand)

Estudamos o modelo dos sapos na árvore homogênea, um sistema de partículas a tempo discreto cuja dinâmica é sintetizada a seguir. No instante inicial, existe em cada vértice da árvore um número aleatório independente e identicamente distribuído de partículas; aquelas posicionadas em um vértice fixado estão ativas, as demais inativas. Partículas ativas realizam passeios aleatórios simples, independentes, a tempo discreto, com probabilidade de desaparecimento (1 - p) em cada instante. Uma partícula inativa torna-se ativa assim que seu vértice é visitado por uma partícula ativa. Consideramos nesta tese o valor crítico p_c que separa a fase em que o processo se extingue quase certamente da fase em que existem partículas ativas em todos os instantes com probabilidade positiva. Provamos um limitante superior para a probabilidade crítica p_c, o qual melhora o resultado anteriormente conhecido para o caso de configuração inicial de uma partícula por vértice. O argumento utilizado consiste na descrição do modelo dos sapos como um modelo de percolação orientada que domina processos de ramificação convenientemente definidos. Obtemos também o valor assintótico do limitante superior estabelecido, mostrando ser igual ao valor assintótico da probabilidade crítica. / We study the frog model on the homogeneous tree, a discrete-time particle system whose dynamics is summarized next. Initially there is an independent and identically distributed random number of particles at each vertex of the tree; those placed at a fixed vertex are active, the others being inactive. Active particles perform independent discrete-time simple random walks, with probability of disappearance (1 - p) at each instant. An inactive particle becomes active once its vertex is hit by an active particle. We consider in this thesis the critical value p_c that separates the phase in which the process dies out almost surely from the phase in which there exist active particles at all times with positive probability. We prove an upper bound for the critical probability p_c, which improves the formerly known result for the case of one particle per vertex initial configuration. The employed argument builds on the description of the frog model as an oriented percolation model which dominates suitably defined branching processes. We also obtain the asymptotic value of the stated upper bound, showing that it equals the asymptotic value of the critical probability.

L'objectif de ce travail consiste à améliorer nos connaissances sur les mécanismes à l'origine des effets de renforcement dans le cas des matériaux nanocomposites. Dans cette optique, des systèmes modèles, à matrice polymère amorphe renforcée par des nanoparticules de silice colloïdale, ont été élaborées. Les propriétés optiques, physico-chimiques et mécaniques de ces matériaux ont été reliés à leur morphologie. L'analyse morphologique des nanocomposites a permis de montrer que les nanoparticules de silice étaient agrégées pour des taux de charges inférieurs à 6%, un chemin percolant apparaissant pour des taux de charges supérieurs. L'étude de la mobilité moléculaire des chaînes du polymère a permis de montrer que les nanoparticules n'induisaient aucune variation de mobilité globale des chaînes. L'augmentation de la partie réelle du module et les variations d'amplitude et de position de la relaxation principale à taux de renfort croissant, révélées lors de l'analyse du comportement viscoélastique linéaire des matériaux, ont été reliées à leur différentes architectures. Ces résultats ont ensuite été confrontés aux résultats théoriques issus de deux approches. A l'échelle mésoscopique, nous avons montré au moyen d'un modèle micromécanique adapté que la présence du réseau percolant de charges pouvait être à l'origine du fort effet de renforcement observé et du décalage de la relaxation principale vers les basses températures pour les composites renforcés par des taux de charges supérieurs au seuil de percolation. A l'échelle atomique, nous avons analysé un système PBMA-silice par dynamique moléculaire. Grâce à ce modèle, nous avons pu analyser l'influence de la présence de la charge sur la mobilité moléculaire du polymère et les propriétés mécaniques des nanocomposites.

L'eau est presente dans le manteau terrestre. Elle peut avoir des effets importants sur la plupart des processus qui s'y deroulent. La distribution a grande echelle de l'eau (sous forme d'ions hydrogene) est approchee a l'aide de modeles numeriques. La prise en compte de la convection mantellique montre que la diffusion de l'eau vers les zones de forte solubilite en eau (la zone de transition) n'est pas assez rapide pour empecher le melange convectif. On predit donc la presence d'une phase fluide dans le manteau inferieur ou la solubilite de l'eau dissoute est tres faible. La phase fluide percole vers le haut sous les effets de la poussee d'Archimede et permet l'accumulation d'eau au sommet du manteau inferieur a la verticale des zones de subduction en depit du mouvement convectif descendant. Contrairement a la diffusion de l'eau dissoute dans les mineraux, la percolation d'une phase fluide semble etre un moyen possible de concentrer de l'eau dans le manteau terrestre. Ce reservoir serait localise au niveau de la frontiere entre la zone de transition et le manteau inferieur (Changement de phase spinelle-postspinelle) au voisinage des zones de subduction.

Dans cette étude nous utilisons une méthode expérimentale novatrice, la mesure du bruit en 1/f, pour approfondir la compréhension des mécanismes de transport dans les compo-sites polymères conducteurs : polyaniline (PANI) - polyméthacrylate de méthyle (PMMA). Une description précise du dispositif utilisé pour ces mesures est présentée. L'influence de la forme de l'échantillon, de la position des contacts électriques et de la méthode de mesure (2 ou 4 contacts) sur le niveau de bruit enregistré est discutée. Puis nous montrons que les composites PANI-PMMA présentent un bruit en 1/f dont l'amplitude S_R, en fonction de la concentration et de la résistance R, suit les lois d'échelle de la percolation avec des exposants critiques kappa=2.19 et w=1.15. L'influence des défauts est testée par l'introduction intentionnelle de coupure dans le réseau conducteur de PANI. La plus forte sensibilité de S_R, comparée à R, est mise en évidence expérimentalement et par simulation numérique.

Le but de cette thèse est d'établir la correspondance entre le diagramme de phase d'un fluide simple et sa structure microscopique en termes d'amas. Les calculs de dynamiques moléculaire et Monte-Carlo que nous avons réalisés sur des systèmes de grandes tailles suggèrent que des amas énergétiquement stables existent dans la phase supercritique, c'est à dire pour des températures supérieures à la température critique. Ces amas qui sont plus froids que le fluide, engendrent une ligne de percolation, caractérisée par les exposants de la percolation aléatoire à trois dimensions. Cette ligne, qui part du point critique thermodynamique, est pratiquement iso-énergétique. Par ailleurs, un amas macroscopique non percolant se forme au passage de la courbe de coexistence liquide-gaz pour des densités inférieures à la densité critique. Cette interprétation géométrique de la condensation renforce l'idée que ces amas sont des objets physiques. Des expériences réalisables dans des systèmes macroscopiques sont proposées pour mettre en évidence la présence de ces amas stables et en particulier l'existence de cette ligne de percolation. Le cas des systèmes de petites tailles est également considéré afin de discuter des expériences de fragmentation de noyaux et agrégats atomiques. En effet, par la simulation de l'expansion d'une gouttelette de fluide, nous avons montré que les amas stables étaient les précurseurs des fragments observés lors d'un processus de fragmentation. Bien que les les comportements critiques soient atténués, les propriétés des amas stables restent valables dans ces petits systèmes. Enfin, l'analyse d'une expérience de fragmentation de noyaux atomiques montre un très bon accord entre les données expérimentales et les prédictions théoriques. Outre son intéret sur le plan fondamental, ce travail pourrait permettre une meilleure compréhension des propriétés de solubilité des fluides supercritiques, utilisés dans de nombreuses applications industrielles.