Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-07-25T11:57:07Z No. of bitstreams: 1 Andrezza Cacione.pdf: 1380246 bytes, checksum: 70140b96b4c0cbf9c9bd395f8542f91e (MD5) / Made available in DSpace on 2018-07-25T11:57:07Z (GMT). No. of bitstreams: 1 Andrezza Cacione.pdf: 1380246 bytes, checksum: 70140b96b4c0cbf9c9bd395f8542f91e (MD5) Previous issue date: 2018-04-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The work in question discusses the usage of probabilistic arguments on the Kinetic Theory of gases by James Clerk Maxwell, more specifically when describing the speed of the gas particles. The reflection on the gas behavior and its constituents develops during the 19th century. In this context, it can be verified the use of probabilist arguments in formulation of this knowledge. This work aims to identify the factors that influenced the use of such probabilistic arguments in the development of the Kinetic Theory of gas by Maxwell, and in which sense his approach differentiates from the existing ones, since the probability was already considered in previous formulations of the same theory. In order to identify the subjects studied by Maxwell and the varieties of reflections to which he was exposed, we searched for information on the calendar of the Cambridge University in the period he frequented the institution, we also researched papers published which he might have had contact with and letters exchanged with family and friends / O trabalho em questão versa sobre a utilização de argumentos probabilísticos na Teoria Cinética dos Gases por James Clerk Maxwell, mais especificamente ao descrever a velocidade das partículas que compõem um gás. A reflexão sobre o comportamento dos gases e de seus constituintes ocorreu ativamente ao longo do século XIX. Nesse contexto, percebe-se a utilização de argumentos probabilísticos na construção desse conhecimento. Este trabalho tem como objetivo identificar fatores que influenciaram o uso de argumentos probabilísticos na construção da Teoria Cinética dos Gases por Maxwell e em que sua abordagem a diferencia das teorias já existentes, uma vez que a probabilidade já era considerada em formulações anteriores para a mesma teoria. A fim de identificar os assuntos estudados por Maxwell e a quais tipos de reflexão ele foi exposto, buscamos informações no calendário da Universidade de Cambridge nos anos em que ele a frequentou, em artigos publicados na época com que de alguma forma ele tenha tido contato e também em cartas trocadas com familiares e amigos

Les travaux de cette thèse s'inscrivent dans le domaine des équations aux dérivées partielles et sont liés à la problématique des changements d'échelle dans le contexte de la cinétique des gaz. En effet, sachant qu'il existe plusieurs niveaux de description pour un gaz, on cherche à relier les différentes échelles associées dans un cadre où une part d'aléa intervient. Dans une première partie, on établit la dérivation rigoureuse de l'équation de Boltzmann linéaire sans cut-off en partant d'un système de particules interagissant via un potentiel à portée infinie en partant d'un équilibre perturbé.La deuxième partie traite du passage d'un modèle BGK stochastique avec champ fort à une loi de conservation scalaire avec forçage stochastique. D'abord, on établit l'existence d'une solution au modèle BGK considéré. Sous une hypothèse additionnelle, on prouve alors la convergence vers une formulation cinétique associée à la loi de conservation avec forçage stochastique.Au cours de la troisième partie, on quantifie dans le cas à vitesses discrètes le défaut de régularité dans les lemmes de moyenne et on établit un lemme de moyenne stochastique dans ce même cas. On applique alors le résultat au cadre de l'approximation de Rosseland pour établir la limite diffusive associée à ce modèle.Enfin, on s'intéresse à l'étude numérique du modèle de Uchiyama de particules carrées à quatre vitesses en dimension deux. Après avoir adapté les méthodes de simulation développées dans le cas des sphères dures, on effectue une étude statistique des limites à différentes échelles de ce modèle. On rejette alors l'hypothèse d'un mouvement Brownien fractionnaire comme limite diffusive / The work of this thesis belongs to the field of partial differential equations and is linked to the problematic of scale changes in the context of kinetic of gas. Indeed, knowing that there exists different scales of description for a gas, we want to link these different associated scales in a context where some randomness acts, in initial data and/or distributed on all the time interval. In a first part, we establish the rigorous derivation of the linear Boltzmann equation without cut-off starting from a particle system interacting via a potential of infinite range starting from a perturbed equilibrium. The second part deals with the passage from a stochastic BGK model with high-field scaling to a scalar conservation law with stochastic forcing. First, we establish the existence of a solution to the considered BGK model. Under an additional assumption, we prove then the convergence to a kinetic formulation associated to the conservation law with stochastic forcing. In the third part, first we quantify in the case of discrete velocities the defect of regularity in the averaging lemmas. Then, we establish a stochastic averaging lemma in that same case. We apply then the result to the context of Rosseland approximation to establish the diffusive limit associated to this model.Finally, we are interested into the numerical study of Uchiyama's model of square particles with four velocities in dimension two. After adapting the methods of simulation which were developed in the case of hard spheres, we carry out a statistical study of the limits at different scales of this model. We reject the hypothesis of a fractional Brownian motion as diffusive limit