Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, a partição iônica e o potencial de membrana em um eritrócito são analisados via equação de Poisson-Boltzmann modificada, considerando as interações não eletrostáticas presentes entre os íons e macromoléculas, assim como, o potencial β. Este potencial é atribuído à diferença de potencial químico de referência entre os meios intracelular e extracelular e ao transporte ativo de íons. O potencial de Gibbs-Donnan via equação de Poisson-Boltzmann na presença de carga fixa em um sistema contendo uma membrana semipermeável também é estudado. O método de aproximação paraboloide em elementos finitos em um sistema estacionário e unidimensionalé aplicado para resolver a equação de Poisson-Boltzmann em coordenadas cartesianas e esféricas. O parâmetro de dispersão relativo às interações não eletrostáticas écalculado via teoria de Lifshitz. Os resultados em relação ao potencial de Gibbs-Donnan mostram-se adequados, podendo ser calculado pela equação de Poisson-Boltzmann. No sistema contendo um eritrócito, quando o potencial β é considerado igual a zero, não se verifica a diferença iônica observada experimentalmente entre os meios intracelular e extracelular. Dessa forma, os potenciais não eletrostáticos calculados via teoria de Lifshitz têm apenas uma pequena influência no que se refere à alta concentração de íon K+ no meio intracelular em relação ao íon Na+ / In this work, the ionic partition and the membrane potential in an erythrocyte are analyzed by modified Poisson-Boltzmann equation, considering non-electrostatic interactions between ions and macromolecules as well as the β potential. This potential is attributed to the difference in chemical potential reference states between intracellular and extracellular environment and the active transport of ions. The Gibbs-Donnan potential is also studied usingthe Poisson-Boltzmann equation with fixed chargeon a system containing a semipermeable membrane. The second order spline finite elements methodin a steady one-dimensional system is applied to solve the Poisson-Boltzmann equation in Cartesian and spherical coordinates.The dispersion parameter of the non-electrostatic interactions is calculated by Lifshitztheory. The results regarding the Gibbs-Donnan potential are adequate and can be calculated by the Poisson-Boltzmann equation. In a system containing an erythrocyte, when the β potential is considered equal to zero, it doesnt check the ionic difference observed experimentally between the intracellular and extracellular environment. Thus, non-electrostatic interactions calculated by Lifshitz theory have only a small influence in the high K+level inside cells while keeping Na+ outside

Sistemas cujos componentes interagem por meio de forças de longo alcance não-blindadas por exemplo, sistemas estelares e plasmas não-neutros têm algumas características anô- malas em relação a sistemas com forças blindadas ou de curto alcance. Além de apresentarem características termodinâmicas peculiares como calor especí co negativo e inequivalência de ensembles, sua dinâmica é predominantemente não-colisional e leva à estados quasiestacion ários fora de equilíbrio. Esses estados são notoriamente difíceis de prever dada uma condição inicial qualquer, e ainda não existe uma teoria uni cada para tratá-los. O equilíbrio termodinâmico é atingido somente após tempos longos que escalam com o tamanho do sistema, muitas vezes excedendo o tempo de vida do universo. A relaxação para o equilíbrio, portanto, tem duas escalas de tempo: uma, curta, que leva a estados quasi-estacionários fora de equilíbrio, e a segunda, longa, que leva ao equilíbrio termodinâmico. Nesta tese de doutorado, examinamos esses fenômenos aplicando modelos teóricos e simulação numérica para diferentes sistemas de interação de longo-alcance, incluindo um modelo de spins clássicos tipo XY com longo alcance, e o sistema auto-gravitante em três dimensões. Em uma segunda etapa, estudamos a relaxação para o equilíbrio termodinâmico, a relaxação colisional, através de equações cinéticas e simulação numérica. Desta forma, buscamos esclarecer os mecanismos por trás dos estados quasi-estacionários e da relaxação colisional. / Systems whose components interact by unscreened long-range forces for example, stellar systems and non-neutral plasmas have characteristics that are anomalous with respect to systems with shielded or short-range forces. Besides presenting unique thermodynamic properties such as negative speci c heat and inequivalence of ensembles, their dynamics is predominantly collisionless and leads to out-of-equilibrium quasi-stationary states. These states are notoriously di cult to predict given an arbitrary initial condition, and there is still no uni ed theory to treat them. Thermodynamic equilibrium is reached only after long timescales that increase with the system size and often exceed the lifetime of the universe. Relaxation to equilibrium, therefore, has two timescales: one short, leading to outof- equilibrium quasi-stationary states, and a second, longer, which leads to thermodynamic equilibrium. In this thesis, we examine these phenomena by applying theoretical models and numerical simulation for di erent long-range interacting systems, including a model of classical XY-type spins with long-range interactions, and the self-gravitating system in three dimensions. In a second stage we study the collisional relaxation to thermodynamic equilibrium through kinetic equations and numerical simulation. We thus seek to clarify the mechanisms behind the quasi-stationary states and collisional relaxation.

Made available in DSpace on 2014-10-09T12:26:15Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:09:38Z (GMT). No. of bitstreams: 0 / Este trabalho apresenta um estudo da Física dos Sistemas dirigidos por Fontes Externas (ADS). É apresentada a definição de alguns parâmetros estáticos e cinéticos da Física do reator que são importantes na avaliação e definição destes sistemas. O objetivo é demonstrar que há diferenças nestes parâmetros quando o sistema está no nível crítico ou subcrítico. Além disso, o trabalho mostra as diferenças observadas nos parâmetros para diferentes modelos de cálculo. São mostradas nesta dissertação duas metodologias de cálculo: Gandini&Salvatores e Dulla e são calculados alguns destes parâmetros utilizando as duas metodologias. O código determinístico de transporte ANISN é utilizado no cálculo destes parâmetros. Numa configuração subcrítica do Reator IPEN/MB-01 dirigido por uma fonte externa de nêutrons são calculados alguns parâmetros físicos. No final do trabalho são apresentadas as conclusões obtidas através destes cálculos. / Dissertacao (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN-CNEN/SP

Abstract The problem of describing a system of many interacting particles is one of the most fundamental questions in physics. One of the central theories used in condensed matter physics to address the problem is the Fermi liquid theory developed by L. D. Landau in the 1956. The theory describes interacting fermions, and can be used to explain transport phenomena of electrons in metals and dynamics of helium three. Even when the theory is not directly applicable, it forms a basis against which other, more sophisticated theories can be compared. this thesis the Fermi liquid theory is applied to 3He-4He-mixtures at temperatures where the bosonic 4He part is superfluid, and the mechanical properties of the system are largely determined by the 3He component, treated as a degenerate normal Fermi liquid. The dynamics of strongly interacting liquid 3He can be described as a collection of quasiparticles, elementary excitations of the system, which interact only weakly. In 3He-4He mixtures the interactions can be continuously tuned by changing the temperature and the concentration of the mixture. The scattering time of quasiparticles depends on temperature, and thus the transition from the hydrodynamic limit of continuous collisions at higher temperatures to the collisionless ballistic limit at low temperatures can be studied. This gives invaluable information on the role of the interactions in the dynamics of the system. In this work, by using the Fermi liquid theory and Boltzmann transport equation, the dynamics of helium mixture disturbed by a mechanical oscillator is described in the full temperature range. The solution necessarily is numeric, but new analytical results in the low temperature limit are obtained as well. The numerical approach enables one to study various boundary conditions thoroughly, and allows application of the theory to a specic geometry. It is shown that in order to explain the experimental observations, it is necessary to take into account the reflection of quasiparticles from the walls of the container. For suitable choice of oscillator frequency and container size, second sound resonances are observed at higher temperatures, while in the ballistic limit quasiparticle interference can be seen. The numerical results are in quantitative agreement with experiments, thus attesting the accuracy of Fermi liquid theory. In particular, the previously observed decrease of inertia of a mechanical oscillator immersed in helium at low temperatures is reproduced in the calculations, and is explained by elasticity of the fluid due to Fermi liquid interactions.

Les progrès techniques réalisés dans le cadre des expériences sur les gaz de fermions ultrafroids ont engendré une émulation particulièrement importante ces dernières années. En effet, ces dispositifs expérimentaux permettent de produire des systèmes gazeux ≪ à la carte ≫, notamment grâce au phénomène de résonances de Feshbach qui permet de contrôler le signe de la longueur de diffusion a par application d'un champ magnétique extérieur. Il est alors possible de générer aussi bien une interaction attractive (a < 0) que répulsive (a > 0). La résonance de Feshbach en elle-même se trouve en a → ±∞, cette limite correspondant à un régime de fortes corrélations entre les particules. De plus, dans la région où a est positive, des états lies moléculaires (bosoniques car formés de deux fermions) peuvent se former. En-dessous d'une certaine température, une phase superfluide peut alors apparaitre, et une transition de phase continue entre l'état bosonique et l'état fermionique peut être observée (BEC-BCS crossover). En fonction de la position dans le diagramme de phases, les modes collectifs possèderont des caractéristiques (fréquence, amortissement) différentes. En ce sens, ils constituent une sonde de l'état de la matière et une connaissance précise de ces modes est par conséquent très importante. Le travail présenté dans cette thèse comporte une caractérisation détaillée de plusieurs modes collectifs dans la phase normale du système atomique. L'étude repose principalement sur l'équation de Boltzmann, que nous résolvons de deux façons différentes. La première consiste à utiliser une méthode des moments ≪ améliorée ≫ (c'est-à-dire d'ordre supérieur). La seconde est numérique et a nécessité l'écriture d'un programme de simulation permettant l'incorporation de tous les effets de milieu (potentiel de champ moyen et section efficace). Une attention toute particulière a été apportée à la mise en place des simulations afin de reproduire le plus fidèlement possible les conditions expérimentales. Les techniques expérimentales permettent également désormais la création de gaz polarisés. Nous présenterons donc dans ce travail une étude de ces gaz utilisant notre programme de simulation (mise en évidence des différents régimes de collision), puis une étude plus théorique ayant pour principal objectif d'établir le diagramme de phase encore méconnu de ces gaz particuliers, et enfin de proposer une méthode de calcul des effets de milieu, les techniques habituelles utilisées pour les gaz non polarisés n'étant plus valables / Technical progress on ultra-cold Fermi gases experiments induced numerous studies for the last few years. Using these experimental setups, it is effectively possible to generate ultra-cold gases with selected properties, in particular through the Feshbach resonances phenomenon. This allows us to set the sign of the scattering length a using an external magnetic field. It is then possible to have an attractive interaction (a < 0) as well as a repulsive one (a > 0). The Feshbach resonance itself is defined for infinite values of a (positive or negative), which corresponds to a strongly interacting regime. Moreover, when a > 0, molecular bound states (bosonic because they are made with two fermionic atoms) can appear. Thus, below a critical temperature, a superfluid phase can emerge and a crossover can be observed (from the BEC to BCS superfluid states). Depending on the position on the phase diagram, frequency and damping of collective modes will be different. This is why the collective modes are good probes of the system phase. A precise extensive knowledge of their characteristics is thus very important. This thesis presents a complete study of some of these collective modes in the normal phase. This work mainly relies on the Boltzmann equation which will be solved in two different ways: firstly, with an improved (higher order) version of the so-called moments method; secondly with a numerical solution that has required to write a numerical code in order to take into account the in-medium effects (mean field potential and in-medium cross section). Particular attention has been paid to numerical simulations in order to reproduce as closely as possible the experimental conditions. Moreover, experimental procedures now allow to create spin unbalanced gases. We have shown in this work a study of these systems using the numerical resolution of the Boltzmann equation. Moreover, we have developed a theoretical approach in order to build the phase diagram of these polarized gases, which is not fully described yet. Finally, we have suggested a method to determine the in-medium effects, with the aim to solve the problem emerging with the usual method used in the balanced case

Dans cette thèse, nous nous sommes intéressés à des écoulements complexes où les régimes hydrodynamique et raréfiés coexistent. On retrouve ce type d'écoulements dans des applications industrielles comme les pompes à vide ou encore les rentrées de capsules spatiales dans l'atmosphère, lorsque la distance entre les molécules de gaz devient si grande que le comportement microscopique des molécules doit être pris en compte. Pour ce faire, nous étudions 2 modèles de l'équation de Boltzmann, le modèle BGK et le modèle ES-BGK. Dans un premier temps, nous développons une nouvelle condition au bord permettant une transition continue de la solution du régime raréfié vers le régime hydrodynamique. Cette nouvelle condition permettant de préserver l'asymptotique vers les équations d'Euler compressible est ensuite incluse dans une méthode de frontière immergée pour traiter, à une précision raisonnable (ordre 2), le cas de solides immergés dans un écoulement, sur grilles cartésiennes. L'utilisation de grillescartésiennes permet une parallélisation aisée du code de simulation numérique afin d'obtenir une réduction considérable du temps de calcul, un des principaux inconvénients des modèles cinétiques. Par la suite, une approche dites aux grilles locales en vitesses est présentée réduisant également le temps de calcul de manière importante (jusqu'à 80%). Des simulations 3D sont également présentées montrant l'efficacité des méthodes. Enfin, le transport passive de particules solides dans un écoulement raréfié est étudié avec l'introduction d'un modèle de type Vlasov couplé au modèle cinétique. Grâce à une résolution basée sur des méthodes de remaillage, la pollution de dispositif optiques embarqués sur des satellites dues à des particules issues de la combustion incomplète dans les moteurs contrôlant d'altitude est étudiée. / This work is devoted to the study of complex flows where hydrodynamic and rarefled regimes coexist. This kind of flows are found in vacuum pumps or hypersonic re-entries of space vehicles where the distance between gas molecules is so large that their microscopicbehaviour differ from the average behaviour of the flow and has be taken into account. We then consider two modelsof the Boltzmann equation viable for such flows: the BGK model dans the ES-BGK model.We first devise a new wall boundary condition ensuring a smooth transition of the solution from the rarefled regime to the hydrodynamic regime. We then describe how this boundary condition (and boundary conditions in general) can be enforced with second order accuracy on an immersed body on Cartesian grids preserving the asymptotic limit towards compressible Euler equations. We exploit the ability of Cartesian grids to massive parallel computations (HPC) to drastically reduce the computational time which is an issue for kinetic models. A new approach considering local velocity grids is then presented showing important gain on the computational time (up to 80%). 3D simulations are also presented showing the efficiency of the methods. Finally, solid particle transport in a rarefied flow is studied. The kinetic model is coupled with a Vlasov-type equation modeling the passive particle transport solved with a method based on remeshing processes. As application, we investigate the realistic test case of the pollution of optical devices carried by satellites due to incompletely burned particles coming from the altitude control thrusters

Dans cette thèse, on étudie principalement l’équation de Boltzmann homogène 3D pour les potentiels durs et les potentiels modérément mous et l’équivalence entre une EDS à sauts et l’EDP correspondante. En particulier, on calcule le spectre multifractal de certains processus stochastiques, on étudie le caractère bien-posé et la propagation du chaos pour l’équation de Boltzmann. Dans le premier chapitre, on étudie les propriétés trajectorielle pathologiques du processus stochastique (Vt)t_0 représentant l’évolution de la vitesse d’une particule typique dans un gaz modélisé par l’équation de Boltzmann pour les potentiels durs ou modérément mous. Nous montrons que ce processus est multifractal et qu’il a un spectre déterministe. Pour les potentiels durs, nous donnons aussi le spectre multifractal du processus $X_t =\int_0^t V_s ds$, représentant l’évolution de la position de la particule typique. Dans le deuxième chapitre, nous étudions l’unicité de la solution faible à l’équation de Boltzmann dans la classe de toutes les solutions mesures, pour les potentiels modérément mous. Ceci nous permet aussi d’obtenir un taux quantitatif de propagation du chaos pour le système de particules de Nanbu. / This thesis mainly studies the 3D homogeneous Boltzmann equation for hard potentials and moderately soft potentials and the equivalence between some jumping SDE and the corresponding PDE. In particular, we compute the multifractal spectrum of some stochastic processes, study the well-posedness and the propagation of chaos for the Boltzmann equation. The purpose of the first chapter is to study the pathwise properties of the stochastic process $(V_t)_{t\geq0}, representing the time-evolution of the velocity of a typical particle in a gas modeled by the Boltzmann equation for hard or moderately potentials. We show that this process is multifractal and has a deterministic spectrum. For hard potentials, we also give the multifractal spectrum of the process $X_t =\int_0^t V_s ds$, representing the time-evolution of the position of the typical particle. The second chapter is devoted to study the uniqueness of the weak solution to the Boltzmann equation in the class of all measure solutions, in the case of moderately soft potentials. This allows us to obtain a quantitive rate of propagation of chaos for Nanbu particle system for this singular interaction. Finally in the third chapter, we extend Figalli’s work [19] to study the relation between some jumping SDE and the corresponding Fokker-Planck equation. We prove that for any weak solution $(ft)_{t\in[0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time-marginals are given by the family $(f_t)_{t\in[0,T]$

Cette thèse comporte trois chapitres indépendants, consacrés à l’étude mathématique de trois problèmes physiques distincts, ayant pour modèles trois équations aux dérivées partielles différentes. Ces équations relèvent plus précisément de la méthode des surfaces de niveau, de la théorie de l’écoulement incompressible des matériaux non newtoniens et de la théorie cinétique des gaz raréfiés. Le premier chapitre de la thèse porte sur la dynamique des frontières en mouvement et contient une justification mathématique de la procédure numérique dite de ré-initialisation, dont les applications sont nombreuses dans le contexte de la célèbre méthode des surfaces de niveau. Nous appliquons ces résultats pour une classe d’équations issues de la méthode des surfaces de niveau de premier ordre. Nous écrivons la procédure de ré-initialisation comme un algorithme de décomposition et nous étudions la convergence de l’algorithme en utilisant des techniques d’homogénéisation dans la variable temporelle. Grâce à cette analyse rigoureuse nous introduisons également une nouvelle méthode pour l’approximation de la fonction de distance dans le contexte de la méthode des surfaces de niveau. Dans le cas où l’on cherche seulement une fonction de l’ensemble de niveau avec un gradient minoré proche du niveau zéro, nous proposons une approximation plus simple. Dans le cas général, où le niveau zéro pourrait présenter des changements de topologie, nous introduisons une nouvelle notion de limites relâchées. Dans le deuxième chapitre de la thèse, nous étudions un problème de frontière libre résultant de l’étude de l’écoulement incompressible d’un matériau non-newtonien, avec limite d’élasticité de type Drucker-Prager, sur un plan incliné et sous l’effet de la pesanteur. Nous obtenons une équation sous-différentielle, que nous formulons comme un problème variationnel avec un terme à croissance linéaire de type gradient, et nous étudions le problème dans un domaine non borné. Nous montrons que les équations sont bien posées et satisfont certaines propriétés de régularité. Nous sommes alors capables de relier les paramètres physiques avec le problème abstrait et de prouver des propriétés quantitatives de la solution. En particulier, nous montrons que la solution a un support compact, la limite de ce que nous appelons la frontière libre. Nous construisons également des solutions explicites d’une équation différentielle ordinaire qui peut estimer la frontière libre. Enfin, le troisième et dernier chapitre de la thèse est dédié aux solutions de l’équation de Boltzmann homogène avec molécules maxwelliennes et énergie infinie. Nous obtenons de nouveaux résultats d’existence de solutions éternelles pour cette équation dans un espace de mesures de probabilité d’énergie infinie (i.e. de moment d’ordre deux infini). Elles permettent de décrire le comportement asymptotique en temps d’autres solutions d’énergie infinie, mais elles apparaissent aussi comme des états asymptotiques intermédiaires dans l’étude des solutions d’énergie finie, mais arbitrairement grande. Les méthodes issues de l’analyse harmonique sont utilisées pour étudier l’équation de Boltzmann, où la variable de vitesse est exprimée en Fourier. Enfin, un changement d’échelle logarithmique en la variable temporelle permet de déterminer le bon comportement asymptotique à l’infini des solutions / This thesis consists of three different and independent chapters, concerning the mathematical study of three distinctive physical problems, which are modelled by three non- linear partial differential equations. These equations concern the level set method, the theory of incompressible flow of non-Newtonian materials and the kinetic theory of rare- fied gases. The first chapter of the thesis concerns the dynamics of moving interfaces and contains a rigorous justification of a numerical procedure called re-initialization, for which there are several applications in the context of the level set method. We apply these results for first order level set equations. We write the re-initialization procedure as a splitting algorithm and study the convergence of the algorithm using homogenization techniques in the time variable. As a result of the rigorous analysis, we are also able to introduce a new method for the approximation of the distance function in the context of the level set method. In the case where one only looks for a level set function with gradient bounded from below near the zero level, we propose a simpler approximation. In the general case where the zero level might present changes of topology we introduce a new notion of relaxed limits. In the second chapter of the thesis, we study a free boundary problem arising in the study of the flow of an incompressible non-Newtonian material with Drucker-Prager plasticity on an inclined plane. We derive a subdifferential equation, which we reformulate as a variational problem containing a term with linear growth in the gradient variable, and we study the problem in an unbounded domain. We show that the equations are well posed and satisfy some regularity properties. We are then able to connect the physical parameters with the abstract problem and prove some quantitative properties of the solution. In particular, we show that the solution has compact support and the support is the free boundary. We also construct explicit solutions of an ordinary differential equation, which we use to estimate the free boundary. The last chapter of the thesis is dedicated to the study of infinite energy solutions of the homogeneous Boltzmann equation with Maxwellian molecules. We obtain new results concerning the existence of eternal solutions in the space of probability measure with infinite energy (i.e. the second order moment is infinite). These solutions describe the asymptotic behaviour of other infinite energy solutions but could also be useful in the study of intermediate asymptotic states of solutions with finite but arbitrarily large energy. We use harmonic analysis tools to study the equation, where the velocity variable is expressed in the Fourier space. Finally, a logarithmic scaling of the time variable allows to determine the correct asymptotic scaling of the solutions

En physique du transport, en particulier en physique du transfert radiatif, la méthode de Monte-Carlo a été développée à l'origine comme la simulation de l'histoire d'un grand nombre de particules, dont on déduit des observables moyennes. Cette méthode numérique doit son succès à plusieurs qualités : une gestion naturelle des espaces des phases aux nombreuses dimensions, une erreur systématique nulle par rapport au modèle physico-mathématique, les intervalles de confiance donnés avec les résultats, une capacité à prendre en compte simultanément de nombreux phénomènes physiques, la possibilité de calcul de sensibilités simultané, et une parallélisation aisée. En cinétique des gaz, les particules collisionnent entre elles et non pas avec un milieu extérieur ; on dit que leur transport est non-linéaire. Ces collisions mutuelles mettent en défaut l'approche évoquée ci-dessus de la méthode de Monte-Carlo ; car pour simuler des trajectoires indépendantes de multiples particules et ainsi estimer leur distribution, il faut connaître au préalable exactement cette même distribution...Cette thèse fait suite à celles de Jérémi DAUCHET (2012) et de Mathieu GALTIER (2014), consacrées au transfert radiatif. Entre autres travaux, ces auteurs montraient comment la méthode de Monte-Carlo peut s'accommoder de non-linéarités, en gardant son formalisme et ses spécificités habituelles. Les non-linéarités alors franchies étaient respectivement une loi de couplage chimie/luminance, et la dépendance de la luminance envers le coefficient d'absorption. On essaie dans ce manuscrit d'outrepasser la non-linéarité du transport. Pour cela, nos principaux outils sont un suivi des particules en remontant le temps, basé sur des formulations intégrales des équations de transport, formulations largement inspirées des algorithmes dits à collisions nulles. Nous montrons, sur plusieurs exemples académiques, que nous avons en effet étendu la méthode de Monte-Carlo à la résolution de l'équation de Boltzmann. Ces exemples sont aussi l'occasion de tester les limites de ce que nous avons mis en place. Les résultats les plus marquants sont certainement l'absence totale de maillage dans la méthode numérique, ainsi que sa capacité à calculer correctement les quantités de particules de haute énergie cinétique (toujours peu nombreuses par rapport au total, en cinétique des gaz). Au-delà des exemples fournis, ce manuscrit est voulu comme un essai de formalisme et une exploration des bases de la méthode développée. L'accent est mis sur les raisonnements menant à la mise au point de la méthode, plutôt que sur les implémentations particulières qui ont été abouties. La méthode est encore, aux yeux de l'auteur, largement susceptible d'être retravaillée. En particulier, les temps maximaux sur lesquels l'évolution des particules est calculable, qui constituent la faiblesse principale de la méthode numérique développée, peuvent sûrement être augmentés. / In transport physics, especially in radiative transfer physics, the Monte-Carlo method has been originally developed as the simulation of the history of numerous particles, from which are deduced mean observables. This numerical method owes its success to several qualities : a natural management of many-dimensional phase space, a null systematic error away from the mathematical and physical model, the confidence intervals given with the results, an ability to take into account simultaneously numerous physical phenomenons, the simultaneous sensitivities calculating possibility, and an easy parallelization. In gas kinetics, particles collide each other, not with an external fixed medium ; it is said that their transport is non-linear. These mutual collisions put out of action the aforesaid approach of the Monte-Carlo method ; because in order to simulate the independent trajectories of multiple particles and thus estimate their distribution, this distribution must beforehand be exactly known...This thesis follows on from those of Jérémy DAUCHET (2012) and of Mathieu GALTIER (2014), dedicated to radiative transfer physics. Between other works, these authors have shown how the Monte-Carlo method can bear non-linearities, while keeping its customary formalism and specificities. The then overcome non-linearities were respectively a chemistry/irradiance coupling law, and the dependence of the irradiance toward the absorption coefficient. We try in this manuscript to overcome the non-linearity of the transport. In this aim, our main tools are a reverse following of particles, based on integral formulations of the transport equations, formulations largely inspired from the so-called null collisions algorithms. We show, on several academic examples, that we have indeed extended the Monte Carlo method to the resolution of the Boltzmann equation. These examples are also occasions to test the limits of what we have built. The most noteworthy results are certainly the absence of any mesh in the numerical method, and its capacity to calculate correctly the high-speed particles quantities (always rare compared to the total, in gas kinetics). Beyond the given examples, this manuscript is wanted as a formalism attempt and an exploration of the developed method basics. The focus is made on the reasoning leading to the method, rather than on particular implementations which have been realized. In the eyes of the author, the method is still largely reworkable. In particular, the maximal times on which the evolution of particles is computable, which constitute the main weakness of the developed numerical method, can surely be increased.

No description available.