We study the application of Lie symmetry methods to solve some cases of the Boltzmann equation, a cornerstone of kinetic theory. The study explores hidden invariances and symmetry-based solutions that help to clarify the complexities inherent in the structure of the equation. Moreover, the study demonstrates a novel approach to solving the equation by rewriting it using the Fourier transform in the velocity variable, which resulted in a non-trivial solution to the Boltzmann equation. The findings not only clarify the mathematical underpinnings of the Boltzmann equation but also enhance our understanding of particle interactions in gases. Overall, this thesis not only enriches the theoretical understanding of integro-differential equations through its rigorous approach but also highlights the efficacy of Lie symmetry methods in unraveling the complexities of fundamental equations in physical sciences.

In this thesis, we study the relationship between the generalized eigenvalue problem (GEP) $Ax=\lambda Bx$, and systems of differential equations. We examine both the Jordan canonical form and Kronecker's canonical form (KCF). The first part of this work provides an introduction to the fundamentals of generalized eigenvalue problems and methods for solving this problem. We discuss the QZ algorithm, which can be used to determine the generalized eigenvalues and also how it can be implemented on MATLAB with the built in function 'eig'.  One essential facet of this work is the exploration of symmetric matrix pencils, which arise when A and B are both symmetric matrices. Furthermore we discuss discrete velocity models (DVMs) focusing specifically on a 12-velocity model on the plane. The results obtained are then applied to half-space problems for discrete velocity models, with a focus on planar stationary systems.

Cette thèse aborde le problème du transport diffusif dans les matériaux multicouches lorsque l'épaisseur des couches est comparable voire plus petit que le libre parcours moyen. Nous présentons un formalisme qui à la fois effectue une synthèse et permet d'aller au delà des divers modèles existants, dérive-diffusion, le modèle Valet-Fert, la méthode des flux ou encore le modèle de Fuchs-Sondheimer. Ce formalisme est applicable à deux types de structures: (i) la géométrie dite CPP (Current Perpendicular to Plane) où le courant moyen est perpendiculaire aux interfaces séparant les couches, et (ii) la géométrie dite CIP (Current In Plane) où le courant moyen est parallèle aux interfaces. Ce nouveau modèle de transport est bâti à partir d'une équation de Boltzmann où les collisions dans les couches et aux interfaces sont représentées par des intégrales de collision linéaires pouvant décrire aussi bien des réflexions spéculaires que des collisions aléatoires non nécessairement isotropes. La résolution de cette équation de Boltzmann pour déterminer les quantités macroscopiques locales d'intérêt se fait en trois étapes : pour chacune des couches, (1) la distribution locale des particules est séparée en deux « chiralités » caractérisés par le signe de la projection du vecteur vitesse de chaque particule le long de l'axe perpendiculaire aux interfaces ; (2) la description locale complète de la distribution angulaire des vitesses pour chaque chiralité est obtenue en développant sur une nouvelle base polynômes orthogonaux adaptée à l'existence de deux chiralités ; (3) pour effectuer la moyenne chirale sur la distribution angulaire des vitesses on définit une troncature minimale de ce développement adaptée aux quantités macroscopiques locales d'intérêt.L’étape (1) est nécessaire afin de pouvoir décrire correctement les collisions d'interfaces, l'étape (3) est usuelle mais l'ingrédient clef de ce formalisme est le point (2) qui seul permet de rendre cohérent les étapes (1) et (3) en présence d'interfaces. Pour la géométrie CPP, ce formalisme « Boltzmann chiral » permet d'unir les systèmes balistique et diffusif sous une même approche macroscopique. En présence de polarisation en spin, ce nouveau formalisme permet d'obtenir entre autre les résistances d'interfaces du modèle Valet-Fert en fonction des coefficients de transmission généralisés associés aux collisions d'interface. Pour les structures CIP, ce modèle permet d'obtenir des expressions analytiques pour les conductivités locales par couche (avec ou sans polarisation en spin) et de plus il rend le lien avec le transport CPP plus transparent. Ce formalisme n'étant pas propre au transport électrique, nous montrons sa versatilité sur une application au transport lumineux en revisitant le problème de Milne pour lequel nous retrouvons un résultat exact de façon beaucoup plus simple. Nous présentons pour terminer une méthode variationnelle fournissant une interprétation intéressante du modèle de Fuchs-Sondheimer. / This thesis addresses the problem of diffusive transport in multilayer systems when the layers thickness is of the order of or even smaller than the mean free path. We present a formalism which enables to synthetize and to go beyond various the standard models (drift-diffusion, Valet-Fert model, flux method or Fuchs-Sondheimer model). This formalism applies to two kinds of structures: (i) the so called CPP geometry (Current Perpendicular to Plane) where the mean transport current is perpendicular to the interfaces separating the layers, and (ii) the so called CIP (Current in Plane) geometry in which the mean transport current is parallel to interfaces. This new model of transport is build on the Boltzmann transport equation in which the scattering in the layer or at interfaces is represented by linear collision integrals that can describe specular and random scattering not necessarily isotropic. The resolution of this Boltzmann equation to obtain macroscopic quantities of interest is done in three steps for each layer: (1) the particle distribution is splitted into two “chiralities” characterized by the sign of the projection of the velocity vector of each particle along the axis perpendicular to interfaces; (2) the local description of the complete angular velocity distribution for each chirality is obtained by an expansion over a new orthogonal polynomial basis adapted to the existence of two chiralities; (3) to compute the chiral mean of the angular velocity distribution we define a minimal troncated expansion adapted to the local physical quantities of interest. Step (1) is necessary to describe correctly the interface scattering, step (3) is usual but the key ingredient of our formalism is step (2) which solely allows a coherent description of step (1) and (3) in the presence of interfaces. For spin polarized systems this novel formalism allows, among other things, to express the boundaries resistances of the Valet-Fert model in terms of generalized transmission coefficients associated to scattering at interfaces. For CIP structures, with this new approach we obtain explicit analytical expressions for the local conductivity of each layer (with or without spin polarisation) and we make the link with CPP transport more transparent. This novel formalism is not specific to electrical transport, to show its versatility we present an application to transport of light by revisiting the Milne problem for which we can recover certain exact result in a much simpler way. At last, we present a variational method which gives some interesting interpretation of the Fuchs-Sondheimer model.

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.

Cette thèse est dédiée à l'étude mathématique et numérique d'une classe d'équations cinétiques collisionnelles, de type équation de Boltzmann. Nous avons porté un intérêt tout particulier à l'équation des milieux (ou gaz) granulaires, initialement introduite dans la littérature physique pour décrire le comportement hors équilibre de matériaux composés d'un grand nombre de grains, ou particules, non nécessairement microscopiques, et interagissant par des collisions dissipant l'énergie cinétique. Ces modèles se sont révélés avoir une structure mathématique très riche. Cette thèse se structure en trois partie pouvant être lues de manière indépendante, mais néanmoins en rapport avec des équations cinétiques collisionnelles en général, et l'équation des milieux granulaires en particulier. La première partie est dédiée à l'étude mathématique du comportement asymptotique de certaines équations cinétiques collisionnelles dans un cadre homogène en espace. Nous y montrons des résultats de type explosion et convergence vers la solution autosimilaire avec calcul explicite des taux, pour des opérateurs de type Boltzmann, grâce à l'utilisation (entre autre) d'une nouvelle méthode de changement de variables dépendant directement de la solution de l'équation considérée. En particulier, nous démontrons que pour un modèle de gaz granulaire - dit anormal - il est possible d'observer une explosion en temps fini. Dans la deuxième partie, orientée analyse numérique et calcul scientifique, nous nous intéressons développement et à l'étude de méthodes spectrales pour la résolution de problèmes multi-échelles, issus de la théorie des équations cinétiques collisionnelles. Les méthodes de changement de variables tiennent aussi une place importante dans cette partie, et permettent d'observer numériquement des phénomènes non triviaux qui apparaissent lors de l'étude de gaz granulaires, comme la création d'amas de matière ou la caractérisation précise du retour vers l'équilibre. La troisième et dernière partie est dédiée à l'étude spectrale de l'opérateur des milieux granulaires avec bain thermique, linéarisé au voisinage d'un équilibre homogène en espace, afin d'établir des résultats de type stabilité et convergence vers une limite hydrodynamique. Ce travail est en fait la généralisation d'un résultat célèbre dans la théorie de l'équation de Boltzmann, dû à R. Ellis et M. Pinsky, et établissant rigoureusement la première limite hydrodynamique vers les équations d'Euler compressibles linéaires puis Navier-Stokes de cette équation / This dissertation is dedicated to the mathematical and numerical study of a class of collisional kinetic equations, such as the Boltzmann equation of perfect gases. We took a particular interest in the granular media (or gases) equation, which has been first introduced in the physical literature to describe the nonnequilibrium behavior of materials composed of a large number of grains (the particles) of macroscopic size, interacting through energy dissipative collisions. These models have a very rich mathematical structure. This dissertation is divided in three independent part, all related to the theory of collisional kinetic equation, with a strong emphasis on granular media. The first part concerns the mathematical study of the asymptotic behavior of space homogeneous Boltzmann-like kinetic equations. We prove some blow up results, as well as convergence towards self-similarity, with explicit rates for two different models. One of the key tools of our proofs is the use of a new scaling method, where the scaling function depends on the solution itself. We especially prove that for a particular model of granular gases (also know as anomalous), finite time blow up occurs. The second part is dedicated to the development and study of spectral methods for the resolution of multi-scale problems, coming from the theory of collisional kinetic equations. Some rescaling methods take a very important place in this part, allowing to observe numerically some nontrivial phenomena such as the clustering in space which occurs in the time evolution of a space inhomogeneous granular gas, or to investigate numerically the trend to equilibrium for this equation. The whole third (and last) part is dedicated to the spectral study of the granular gases operator with a thermal bath, linearized near a space homogeneous self-similar profile. The goal of this work is to prove some stability results for the complete space inhomogeneous equation, and to investigate the hydrodynamic limit of the model. This work is based and extend the famous result of R. Ellis and M. Pinsky on the spectrum of the linearized Boltzmann equation, intended to establish rigorously the hydrodynamic limit of this equation towards the linearized Euler and Navier-Stokes equations

Grace aux progrès énormes des techniques de refroidissement, des expériences actuelles avec des atomes fermioniques piégés atteignent des températures extrêmement basses de l'ordre du nanoKelvin. Le but principal de ces expériences est l'étude de la transition nommée "BEC-BCS crossover". Pour cela, on change le champ magnétique autour d'une résonance de Feschbach, ce qui implique que la longueur de diffusion change des valeurs répulsives (a positif), à travers la limite unitaire (a infini) aux valeurs attractives (a négatif). Du côté BEC, où le système forme un condensat de Bose-Einstein de molécules fortement liées, aussi bien que du côté BCS, où les atomes forment des paires de Cooper qui ont une grande extension par rapport à la distance moyenne entre les atomes, on s'attend à ce que le système devienne superfluide, à condition que la température soit inférieure à une certaine température critique. Afin de trouver des signes sans équivoque de la superfluidité, il est nécessaire de regarder des observables dynamiques comme l'expansion du nuage atomique lorsque le piège est éteint ou des oscillations collectives du nuage. Le travail effectué au cours de cette thèse est une étude de la dynamique des modes collectifs dans les gaz de fermions froids. Nous avons développé un modèle basé sur l'évaluation de la matrice T. L'utilisation de l'équation de transport de Boltzmann pour les particules permet ensuite une étude semi-numérique des modes collectifs dans tous les régimes d'interaction. Cette étude a permis de mettre en évidence pour la première fois que la fréquence du mode radial quadrupolaire est supérieure à deux fois la fréquence du piège, comme cela est vérifié expérimentalement et contrairement aux premières théories n'incluant pas les effets de champ moyen. Les résultats obtenus ont aussi mis en évidence la nécessité d'une résolution numérique complète de l'équation de Boltzmann et de l'amélioration des techniques de détermination des observables physiques du gaz. Cette résolution numérique de l'équation de Boltzmann a montré que la détermination du temps de relaxation par la méthode des moments est erronée de 30%, ce qui influe fortement sur la détermination de la fréquence et de l'amortissement du mode collectif. Enfin, l'amélioration de la méthode des moments, considérant l'ordre supérieur, permet d'améliorer sensiblement l'accord avec le résultat numérique. Une telle investigation n'avait jamais été réalisée et montre la nécessité de considérer les moments d'ordre supérieurs pour l'étude des modes collectifs par l'équation de Boltzmann d'un gaz de fermions dans la phase normale / Due to the improvement of cooling technics, recent experiments on ultracold Fermi gases now reach temperatures in the range of the nanokelvin. The main goal of these experiments is to study the so called BEC-BCS crossover. By tuning the external magnetic field around a Feschbach resonance, the scattering lengh runs from the repulsive side (a>0) to the attractive side (a<0) through the unitarity limit (a infinite). In the BEC side, where the system can form a Bose Einstein condensate of strongly bounded molecules and also in the BCS side where the atoms can form Cooper pairs, the system can become superfluid if the temperature is below the critical temperature. In order to know if the system is superfluid, one has to look at dynamical observables such the collective oscillations of the cloud. The work of this thesis deals with the dynamics of collective modes of ultracold Fermi gases. We have developped a model based on the T-matrix evaluation. We use the Boltzmann transport equation to study the collective modes in all the interaction regimes. This work shows for the first time that the frequency of the radial quadrupole mode can be more than twice the trap frequency, as it has been observed by experiments. The result comes from the incorporation of medium effects. These first results have also shown the necessity to solve numerically the Boltzmann equation. The numerical resolution shows that the determination of the relaxation time with the methods of moments was wrong by a factor 30%. Consequently, the determination of the frequency and the damping of the collective mode by this analytical method is also wrong. Thus, the improvement of this method, by considering the higher order, provides a better agreement with the numerical results. Such a calculation has never been done and shows the necessity to consider the higher order moments in the description of the collective mode of an ultracold Fermi gas in the normal with the Boltzmann equation

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 &#946;. 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 &#946; 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 &#946; 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 &#946; 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

Ce travail est dédié à la simulation d’écoulements multidimensionnels de gaz raréfiés dans un domaine où l’interface avec le solide est mobile. Le comportement du gaz est modélisé par un modèle de type BGK de l’équation de Boltzmann et une méthode déterministe de vitesses discrètes est utilisée pour discrétiser l’espace des vitesses microscopiques.Dans ce document, nous proposons tout d’abord trois discrétisations spatiales du modèle qui permettent la prise en compte du mouvement des parois solides, grâce à un traitement spécifique des conditions aux limites. Ces approches sont implémentées et validées pour plusieurs cas unidimensionnels et à la suite de cette étude, la méthode maille coupée est choisie pour une extension à des écoulements de dimensions plus élevées.La suite du travail présente l’algorithme utilisé pour la simulation d’écoulements 2D et 3D. La précision et la robustesse de l’implémentation sont mises en avant grâce à la simulation de nombreux cas tests, dont les résultats sont comparés à ceux issus de la littérature. La méthode maille coupée a notamment été optimisée par une technique de raffinement de maillage adaptatif. La simulation instationnaire 3D de la rotation des pâles du radiomètre de Crookes illustre pleinement le potentiel de la méthode. / This work is devoted to the multidimentional simulation of rarefied gases in a domain with moving boundary. The governing equation is given by BGKtype model of Boltzmann equation and velocity space is discretized with a standard discrete velocity method.We first propose three space discretizations that take boundary motion into account by specific treatment of the boundary conditions. These approaches are implemented and validated for several 1D flows. Based on this study, the cut cell method is chosen to be extend to multidimentional flows.Then we detail the cut cell algorithm for 2D and 3D flow simulations. Robustness and accuracy of the implementation are investigated through the simulation of numerous test cases. Our results are rigorously compared to the ones coming from the literature and good agreement is shown. The cut cell method has been optimized with an adaptive refinement mesh technique. The 3D unstationary simulation of the Crookes radiometer rotating vanes is a perfect illustration of the method potential.

<p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p>

The Vlasov-Poisson system is most commonly used to model the movement of charged particles in a plasma or of stars in a galaxy. It consists of a kinetic equation known as the Vlasov equation coupled with a force determined by the Poisson equation. The system in Euclidean space is well-known and has been extensively studied under various assumptions. In this paper, we derive the Vlasov-Poisson equations assuming the particles exist only on the 2-sphere, then take an in-depth look at particles which initially lie along a great circle of the sphere. We show that any great circle is an invariant set of the equations of motion and prove that the total energy, number of particles, and entropy of the system are conserved for circular initial distributions. / Graduate