Stochastic models describing how interacting individuals give rise to collective behaviour have become a widely used tool across disciplines—ranging from biology to physics to social sciences. Continuum population-level models based on partial differential equations for the population density can be a very useful tool (when, for large systems, particle-based models become computationally intractable), but the challenge is to predict the correct macroscopic description of the key attributes at the particle level (such as interactions between individuals and evolution rules). In this thesis we consider the simple class of models consisting of diffusive particles with short-range interactions. It is relevant to many applications, such as colloidal systems and granular gases, and also for more complex systems such as diffusion through ion channels, biological cell populations and animal swarms. To derive the macroscopic model of such systems, previous studies have used ad hoc closure approximations, often generating errors. Instead, we provide a new systematic method based on matched asymptotic expansions to establish the link between the individual- and the population-level models. We begin by deriving the population-level model of a system of identical Brownian hard spheres. The result is a nonlinear diffusion equation for the one-particle density function with excluded-volume effects enhancing the overall collective diffusion rate. We then expand this core problem in several directions. First, for a system with two types of particles (two species) we obtain a nonlinear cross-diffusion model. This model captures both alternative notions of diffusion, the collective diffusion and the self-diffusion, and can be used to study diffusion through obstacles. Second, we study the diffusion of finite-size particles through confined domains such as a narrow channel or a Hele–Shaw cell. In this case the macroscopic model depends on a confinement parameter and interpolates between severe confinement (e.g., a single- file diffusion in the narrow channel case) and an unconfined situation. Finally, the analysis for diffusive soft spheres, particles with soft-core repulsive potentials, yields an interaction-dependent non-linear term in the diffusion equation.

Koagulation ist physikalisch bedeutsam für eine Vielzahl von technischen und naturwissenschaftlichen Anwendungen und bezeichnet die paarweise Verschmelzung von Clustern unterschiedlicher Masse. Der zeitliche Verlauf der Clusterkonzentration läßt sich durch Smoluchowskis Koagulationsgleichung beschreiben, einem unendliches System nichtlinearer Differentialgleichungen. Ausgangspunkt dieser Arbeit ist eine nichtlineare maßwertige Gleichung, die die Koagulations- und andere kinetische Gleichungen beinhaltet und verschiedene physikalische und chemische Mechanismen integriert. Sie ermöglicht einen allgemeinen Zugang zu Fragen bezüglich der Existenz von Lösungen und ihrer Approximation durch stochastische Partikelsysteme. Die Teilchensysteme werden dabei als reguläre Sprungprozesse modelliert, welche eine Menge diskreter Maße auf einem lokal-kompakten Raum als Zustandsraum besitzen. Die Arbeit untergliedert sich in drei Teile: Unter geeigneten Voraussetzungen an die Sprungraten werden zunächst für wachsende Teilchenzahlen Approximations- und Konvergenzaussagen unter Verwendung von Kompaktheitsargumenten, Martingaltheoremen und Lokalisierungstechniken bewiesen. Ihre Anwendung auf die Koagulationsgleichung mit Fragmentation, Quellen und Senken erlaubt anschließend die Herleitung neuer Existenzresultate und stochastischer Algorithmen. Der letzte Abschnitt illustriert die numerischen Eigenschaften und die Effizienz der neuen Algorithmen im Vergleich zu bisherigen Monte Carlo Methoden und ihre besondere Eignung zur Analyse des Gelationsphänomens, einem Phasenübergang, welcher zum Masseverlust im Clustersystem führt. / Coagulation is an important physical process for a wide range of technical and scientific applications and denotes the pairwise merging of clusters with different mass. The dynamic behaviour of the cluster concentration can be described by Smoluchowski's coagulation equation which is an infinite system of nonlinear differential equations. In this thesis we start with a nonlinear measure-valued equation generalizing the coagulation and other kinetic equations and integrating various physical and chemical processes. This equation allows a unified treatment of questions concerning existence of solutions and their approximation by means of stochastic particle systems. Here, the particle systems are defined as regular jump processes living on a set of point measures on a locally compact space. The thesis consists of three parts: First of all, approximation and convergence results for suitable jump rates and increasing particle numbers are proved by means of compactness theorems, martingale techniques and localizing procedures. Then, an application to the coagulation equation with fragmentation, source and efflux terms leads to new existence results and stochastic algorithms. Finally, their numerical features and efficiency are compared to known Monte Carlo methods and their specific convergence properties are presented with respect to a phase transition which is called gelation and leads to a loss of total cluster mass.

Nessa tese, estudamos o limite hidrodinâmico de um sistema estocástico de neurônios cujas interações são dadas por potenciais de Kac que imitam sinapses elétricas e químicas, e as correntes de vazamento. Esse sistema consiste de $\\ep^$ neurônios imersos em $[0,1)^2$, cada um disparando aleatoriamente de acordo com um processo pontual com taxa que depende tanto do seu potential de membrana como da posição. Quando o neurônio $i$ dispara, seu potential de membrana é resetado para $0$, enquanto que o potencial de membrana do neurônio $j$ é aumentado por um valor positivo $\\ep^2 a(i,j)$, se $i$ influencia $j$. Além disso, entre disparos consecutivos, o sistema segue uma movimento determinístico devido às sinapses elétricas e às correntes de vazamento. As sinapses elétricas estão envolvidas na sincronização do potencial de membrana dos neurônios, enquanto que as correntes de vazamento inibem a atividade de todos os neurônios, atraindo simultaneamente todos os potenciais de membrana para $0$. No principal resultado dessa tese, mostramos que a distribuição empírica dos potenciais de membrana converge, quando o parâmetro $\\ep$ tende à 0 , para uma densidade de probabilidade $ho_t(u,r)$ que satisfaz uma equação diferencial parcial nâo linear do tipo hiperbólica . / We study the hydrodynamic limit of a stochastic system of neurons whose interactions are given by Kac Potentials that mimic chemical and electrical synapses and leak currents. The system consists of $\\ep^$ neurons embedded in $[0,1)^2$, each spiking randomly according to a point process with rate depending on both its membrane potential and position. When neuron $i$ spikes, its membrane potential is reset to $0$ while the membrane potential of $j$ is increased by a positive value $\\ep^2 a(i,j)$, if $i$ influences $j$. Furthermore, between consecutive spikes, the system follows a deterministic motion due both to electrical synapses and leak currents. The electrical synapses are involved in the synchronization of the membrane potentials of the neurons, while the leak currents inhibit the activity of all neurons, attracting simultaneously their membrane potentials to 0. We show that the empirical distribution of the membrane potentials converges, as $\\ep$ vanishes, to a probability density $ho_t(u,r)$ which is proved to obey a nonlinear PDE of Hyperbolic type.

Cette thèse se compose de plusieurs travaux portant sur deux branches de la théorie des probabilités: processus de particules et cartes planaires aléatoires. Un premier travail concerne les aspects algébriques des mesures invariantes des processus de particules. Nous obtenons des conditions nécessaires et suffisantes sous lesquelles un processus de particules en temps continu avec espace d’états local discret possède une mesure invariante simple. Dans un deuxième travail nous étudions un modèle "biologique" de coexistence de 2 espèces en compétition sur un espace partagé, et soumis à des épidémies modélisées par un modèle probabiliste appelé "feux de forêts". Notre résultat principal montre que pour deux espèces, il existe des régions explicites de paramètres pour lesquelles une espèce domine ou les deux espèces coexistent. Il s’agit d’un des premiers modèles pour lesquels la coexistence d’espèces sur le long terme est prouvée. Les troisièmes et quatrièmes travaux. portent sur les cartes planaires décorées par des arbres. Dans le troisième nous présentons une bijection entre l’ensemble des cartes décorées par des arbres et le produit Cartésien entre l’ensemble des arbres planaires et l’ensemble de cartes à bord simple. Nous obtenons quelques formules de comptage et quelques outils pour l’étude de cartes aléatoires décorées par un arbre. Le quatrième travail montre que les triangulations et quadrangulations aléatoires uniformes avec f faces, bord simple de taille p et décorées par un arbre avec a arêtes, convergent en loi pour la topologie locale vers différentes limites, dépendant du comportement fini ou infini de la limite de f, p et a. / This thesis consists in several works exploring some models belonging to two branches of probability theory: interacting particle systems and random planar maps. A first work concerns algebraic aspects of interacting particle systems invariant measures. We obtain some necessary and sufficient conditions for some continuous time particle systems with discrete local state space, to have a simple invariant measure. In a second work we investigate the effect on survival and coexistence of introducing forest fire epidemics to a certain two-species spatial competition model. Our main results show that, for the two-type model, there are explicit parameter regions where either one species dominates or there is coexistence; contrary to the same model without forest fires, for which the fittest species alwaysdominates. The third and fourth works are related to tree-decorated planar maps. In the third work we present a bijection between the set of tree-decorated maps and the Cartesian product between the set of trees and the set of maps with a simple boundary. We obtain some counting results and some tools to study random decorated map models. In the fourth work we prove that uniform tree-decorated triangulations and quadrangulations with f faces, boundary of length p and decorated by a tree of size a converge weakly for the local topology to different limits, depending on the finite or infinite behavior of f, p and a.

Nanometer-scale particles of the noble metals have been used for decades as contrast enhancement agents in electron microscopy. Over the past several years it has been demonstrated that these particles also function as excellent contrast agents for optical imaging techniques. The resonant optical scattering they exhibit enables scattering cross sections that may be many orders of magnitude greater than the analogous efficiency factor for fluorescent dye molecules. Biologically relevant labeling with nanoparticles generally results in aggregates containing a few to several tens of particles. The electrodynamic coupling between particles in these aggregates produces observable shifts in the resonance-scattering spectrum. This dissertation provides a theoretical analysis of the scattering from nanoparticle aggregates. The key objectives are to describe this scattering behavior qualitatively and to provide numerical codes usable for modeling its application to biomedical engineering. Considerations of the lowest-order dipole-dipole coupling lead to simple qualitative predictions of the behavior of the spectral properties of the optical cross sections as they depend on number of particles, inter-particle spacing, and aggregate aspect ratio. More comprehensive analysis using the multiple-particle T-matrix formalism allows the elaboration of more subtle cross-section spectral features depending on the interactions of the electrodynamic collective-modes of the aggregate, of individual-particle modes, and of modes associated with groups of particles within the aggregate sub-structure. In combination these analyses and the supporting numerical code-base provide a unified electrodynamic approach which facilitates interpretation of experimental cross section spectra, guides the design of new biophysical experiments using nanoparticle aggregates, and enables optimal fabrication of nanoparticle structures for biophysical applications. / text

In this study, an appearance reconstruction method based on extraction of material re&amp / #64258 / ectance properties of a three-dimensional (3D) object from its twodimensional (2D) images is explained. One of the main advantages of this system is that the reconstructed object can be rendered in real-time with photorealistic quality in varying illumination conditions. Bidirectional Re&amp / #64258 / ectance Distribution Functions (BRDFs) are used in representing the re&amp / #64258 / ectance of the object. The re&amp / #64258 / ectance of the object is decomposed into di&amp / #64256 / use and specular components and each component is estimated seperately. While estimating the di&amp / #64256 / use components, illumination-invariant images of the object are computed from the input images, and a global texture of the object is extracted from these images by using surface particles. The specular re&amp / #64258 / ectance data are collected from the residual images obtained by taking di&amp / #64256 / erence between the input images and corresponding illumination-invariant images, and a Lafortune BRDF model is &amp / #64257 / tted to these data. At the rendering phase, the di&amp / #64256 / use and specular components are blended into each other to achieve a photorealistic appearance of the reconstructed object.

Le processus de contact est l'un des systèmes de particules en interaction les plus étudiés. Il peut s'interpréter comme un modèlepour la propagation d'un virus dans une population ou sur un réseau. L'objectif de cette thèse est d'étudier la relation entre la structure locale du réseau et le comportement global du processus sur le réseau tout entier.Le cadre typique dans lequel on se place est celui d’une suite de graphes aléatoires $(G_n)$ convergeant localement vers un graphe limite $G$.On étudie alors le comportement asymptotique du temps d’extinction $tau_n$ du processussur $G_n$; lorsqu’initialement tous les individus sont infectés. Nous montrons sur plusieurs exemples qu’il existe unetransition de phase lorsque $lambda$ - le taux d'infection du processus - traverse une valeur critique $ lambda_c (G)$, qui ne dépend que de $G$.Plus précisément, pour certains modèles de graphes aléatoires comme le modèle de configuration, le graphe d'attachement préférentiel, le graphe géométrique aléatoire, le graphe inhomogène, nous montrons que $ tau_n $ est d'ordre soit logarithmique soit exponentiel; selon que $ lambda$ est soit inférieur ou supérieur à $lambda_c (G) $.De plus, dans certains cas, nous montrons des résultats de métastablité: en régime sur-critique, $ tau_n $ divisé par son espérance converge en loi vers une variable aléatoire exponentielle de moyenne $1$, et la densité des sites infectés reste stable (et non nulle) sur une période de temps d’ordre typiquement $tau_n$. / The contact process is one of the most studied interacting particle systems and is also often interpreted as a model for the spread of a virus in a population or a network. The aim of this thesis is to study the relationship of the local structure of the network and the global behavior of the contact process (the virus) on the whole network. Let $(G_n)$ be a sequence of random graphs converging weakly to a graph $G$. Then we study $tau_n$, the extinction time of the contact process on $G_n$ starting from full occupancy. We prove in some examples that there is a phase transition of $tau_n$ when $lambda$ - the infection rate of the contact process crosses a critical value $lambda_c(G)$ depending only on $G$. More precisely, for some models of random graphs, such as the configuration model, preferential attachment graph, random geometric graph, inhomogeneous graph, we show that $tau_n$ is of logarithmic (resp. exponential) order when $lambda < lambda_c(G)$ (resp. $lambda < lambda_c(G)$). Moreover, in some cases we also prove metastable results: in the super-critical regime, $tau_n$ divided by its expectation converges in law to an exponential random variable with mean $1$, and the density of the infected sites is stable for a long time.

Nessa tese, estudamos o limite hidrodinâmico de um sistema estocástico de neurônios cujas interações são dadas por potenciais de Kac que imitam sinapses elétricas e químicas, e as correntes de vazamento. Esse sistema consiste de $\\ep^$ neurônios imersos em $[0,1)^2$, cada um disparando aleatoriamente de acordo com um processo pontual com taxa que depende tanto do seu potential de membrana como da posição. Quando o neurônio $i$ dispara, seu potential de membrana é resetado para $0$, enquanto que o potencial de membrana do neurônio $j$ é aumentado por um valor positivo $\\ep^2 a(i,j)$, se $i$ influencia $j$. Além disso, entre disparos consecutivos, o sistema segue uma movimento determinístico devido às sinapses elétricas e às correntes de vazamento. As sinapses elétricas estão envolvidas na sincronização do potencial de membrana dos neurônios, enquanto que as correntes de vazamento inibem a atividade de todos os neurônios, atraindo simultaneamente todos os potenciais de membrana para $0$. No principal resultado dessa tese, mostramos que a distribuição empírica dos potenciais de membrana converge, quando o parâmetro $\\ep$ tende à 0 , para uma densidade de probabilidade $ho_t(u,r)$ que satisfaz uma equação diferencial parcial nâo linear do tipo hiperbólica . / We study the hydrodynamic limit of a stochastic system of neurons whose interactions are given by Kac Potentials that mimic chemical and electrical synapses and leak currents. The system consists of $\\ep^$ neurons embedded in $[0,1)^2$, each spiking randomly according to a point process with rate depending on both its membrane potential and position. When neuron $i$ spikes, its membrane potential is reset to $0$ while the membrane potential of $j$ is increased by a positive value $\\ep^2 a(i,j)$, if $i$ influences $j$. Furthermore, between consecutive spikes, the system follows a deterministic motion due both to electrical synapses and leak currents. The electrical synapses are involved in the synchronization of the membrane potentials of the neurons, while the leak currents inhibit the activity of all neurons, attracting simultaneously their membrane potentials to 0. We show that the empirical distribution of the membrane potentials converges, as $\\ep$ vanishes, to a probability density $ho_t(u,r)$ which is proved to obey a nonlinear PDE of Hyperbolic type.

Particle simulations in real-time become reality only a few years before, when in computer science occured the idea of GPGPU. This new technology allows use the massive force of graphics card for general purposes. Today, the trend is to accelerate existing algorithms by rewriting into parallel form. On this priciple operate the particle systems too. An interesting area of particle systems are fluid simulations. The simulations are based on the theory of Navier-Stokes equations and their numerical solutions with SPH (Smoothed particle hydrodynamics). Liquids are part of everyday life, and therefore it is important to render them realistically. They are used in modern computer games and different visualizations that run in real time, therefore they must be quickly displayed.

The goal of this thesis is to investigate the possibilities of the OGRE visualization library and to implement an application which utilizes both simple and advanced OGRE features. Furhtermore there are explained basics of creating effects using Cg language and connecting them with OGRE library. The Bullet library which is often used in visualization applications for physics calculations is analyzed as well. The second part describes implementation of the demo application. In the end, the results are discussed, including possibilities of future development.