The subject of the thesis is the derivation of non-local diffusion equations from kinetic models with heavy-tailed equilibrium in velocity. We are particularly interested in confining the kinetic equations and developing methods that allow us, from the confined kinetic models, to derive confined versions of non-local diffusion equations.

abstract: Need-based transfers (NBTs) are a form of risk-pooling in which binary welfare exchanges occur to preserve the viable participation of individuals in an economy, e.g. reciprocal gifting of cattle among East African herders or food sharing among vampire bats. With the broad goal of better understanding the mathematics of such binary welfare and risk pooling, agent-based simulations are conducted to explore socially optimal transfer policies and sharing network structures, kinetic exchange models that utilize tools from the kinetic theory of gas dynamics are utilized to characterize the wealth distribution of an NBT economy, and a variant of repeated prisoner’s dilemma is analyzed to determine whether and why individuals would participate in such a system of reciprocal altruism. From agent-based simulation and kinetic exchange models, it is found that regressive NBT wealth redistribution acts as a cutting stock optimization heuristic that most efficiently matches deficits to surpluses to improve short-term survival; however, progressive redistribution leads to a wealth distribution that is more stable in volatile environments and therefore is optimal for long-term survival. Homogeneous sharing networks with low variance in degree are found to be ideal for maintaining community viability as the burden and benefit of NBTs is equally shared. Also, phrasing NBTs as a survivor’s dilemma reveals parameter regions where the repeated game becomes equivalent to a stag hunt or harmony game, and thus where cooperation is evolutionarily stable. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2018

Made available in DSpace on 2014-06-11T19:30:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-03-15Bitstream added on 2014-06-13T21:00:47Z : No. of bitstreams: 1 nova_cv_me_bauru.pdf: 1980542 bytes, checksum: 9aad1da39bf5c30c6e4d38ca0be07fef (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / A teoria cinética dos gases tem como primeira aproximação o comportamento observado considerando o gás como tendo um comportamento ideal, isto é, pode ser modelado através da lei do gás ideal. As equações de estado conhecidas, como a equação do gás ideal e de van der Waals descrevem, dentro das aproximações do modelo, situações bastante diversas. Neste trabalho iremos utilizar a lei dos gases ideais, ou mais especificamente a implementação da Equação de van der Waals para o entendimento do fenômeno de colisões que entre átomos e moléculas / The kinetic theory of gases has a a first approximation the observed behavior considering the gas to have an ideal behavior, ie it can be modeled by the ideal bas law. The equations of state known as the ideal gas equation and Van der Waals describe, within the approximations of the model, very different situations. In this paper we use the ideal gas law, or more specifically the implementation of the Vann der Waals equation for understandign the phenomenon of collisions between atoms and molecules

Cette thèse porte sur le développement de modèles multi-échelles et de méthodes de calcul pour les applications aérothermodynamiques. Le travail de recherche sur les modèles multi-échelles met l’accent sur l’excitation énergétique et la dissociation. L’objectif était double : mieux comprendre la dynamique des processus d'excitation énergétique et dissociation et développer des modèles réduits en diminuant la résolution d’un modèle détaillé de collisions rovibrationnelles. Les résultats obtenus ont montré que les modèles réduits permettent de reproduire avec précision la dynamique d’écoulement prédites par le modèle détaillé de collisions rovibrationnelles. Le travail de recherche sur les méthodes de calcul a porté sur les écoulements raréfiés. L’objectif était de formuler une méthode numérique de type déterministe pour résoudre l’équation de Boltzmann dans le cas de gaz à plusieurs composants y compris l’énergie interne. La méthode numérique est basée sur la structure de convolution pondérée de la transformée de Fourier de l’équation de Boltzmann. La précision de la méthode numérique proposée a été évaluée en comparant les moments extraits de la fonction de distribution de vitesse avec les prédictions de la méthode de simulation directe Monte Carlo (DSMC). Dans toutes les applications étudiées, un excellent accord a été trouvé. / This thesis aimed at developing multi-scale models and computational methods for aerother-modynamics applications. The research on multi-scale models has focused on internal energy excitation and dissociation of molecular gases in atmospheric entry flows. The scope was two-fold: to gain insight into the dynamics of internal energy excitation and dissociation in the hydrodynamic regime and to develop reduced models for Computational Fluid Dynamics applications. The reduced models have been constructed by coarsening the resolution of a detailed rovibrational collisional model developed based on ab-initio data for the N2 (1Σ+g)-N (4Su) system provided by the Computational Quantum Chemistry Group at NASA Ames Research Center. Different mechanism reduction techniques have been proposed. Their appli-cation led to the formulation of conventional macroscopic multi-temperature models and vi-brational collisional models, and innovative energy bin models. The accuracy of the reduced models has been assessed by means of a systematic comparison with the predictions of the detailed rovibrational collisional model. Applications considered are inviscid flows behind normal shock waves, within converging-diverging nozzles and around axisymmetric bodies, and viscous flows along the stagnation-line of blunt bodies. The detailed rovibrational colli-sional model and the reduced models have been coupled to two flow solvers developed from scratch in FORTRAN 90 programming language (SHOCKING_F90 and SOLV-ER_FVMCC_F90). The results obtained have shown that the innovative energy bin models are able to reproduce the flow dynamics predicted by the detailed rovibrational collisional model with a noticeable benefit in terms of computing time. The energy bin models are also more accurate than the conventional multi-temperature and vibrational collisional models. The research on computational methods has focused on rarefied flows. The scope was to formu-late a deterministic numerical method for solving the Boltzmann equation in the case of multi-component gases with internal energy by accounting for both elastic and inelastic collisions. The numerical method, based on the weighted convolution structure of the Fourier trans-formed Boltzmann equation, is an extension of an existing spectral-Lagrangian method, valid for a mono-component gas without internal energy. During the development of the method, particular attention has been devoted to ensure the conservation of mass, momentum and en-ergy while evaluating the collision operators. Conservation is enforced through the solution of constrained optimization problems, formulated in a consistent manner with the collisional in-variants. The extended spectral-Lagrangian method has been implemented in a parallel com-putational tool (best; Boltzmann Equation Spectral Solver) written in C programming lan-guage. Applications considered are the time-evolution of an isochoric gaseous system initially set in a non-equilibrium state and the steady flow across a normal shock wave. The accuracy of the proposed numerical method has been assessed by comparing the moments extracted from the velocity distribution function with Direct Simulation Monte Carlo (DSMC) method predictions. In all the cases, an excellent agreement has been found. The computational results obtained for both space homogeneous and space inhomogeneous problems have also shown that the enforcement of conservation is mandatory for obtaining accurate numerical solutions.

Desenvolvemos urna teoria cinética para urna mistura de gases ionizados em presença de campos elétricos e magnéticos. As leis de Ohm, Fourier e Navier-Stokes são obtidas por dois métodos distintos que se baseiam na equação de Boltzmann. Verificamos que o emprego de teoremas de representação torna o método de Chapman-Enskog mais direto. Entretanto o método combinado mostrou-se extremamente simples, onde os coeficientes de transporte são determinados através da inversão de tensores de segunda e quarta ordens. Calculamos também a integral de colisão para as possíveis interações em gases ionizados tais como, entre partículas carregadas, partícula carregada e partícula neutra e entre partículas neutras. Como uma aplicação do método combinado, determinamos os coeficientes de condutividade elétrica, condutividade térmica, coeficiente termo-elétrico e o coeficiente de viscosidade cisalhante para um gás totalmente ionizado. Obtemos seus respectivos gráficos, considerando então um gás ionizado formado a partir do gás de hélio. / We develop a kinetic theory for ionized gases mixtures under the presence of electric and magnetic fields. The laws of Ohm, Fourier and Navier-Stokes are obtained by two different methods based on the Boltzmann equation. We verify that the use of representation theorems makes the Chapman-Enskog method more direct. However the combined method shows up as extremely simple where the transport coefficients are determined through inversion of second-order and fourth order tensors. We calculate also the collision integrals for possible interactions in ionized gases like: between charged particles, between charged particles and neutral particles and between neutral particles. As an application of the combined method, we determine the electrical and thermal conductivity coefficients, thermo-electric and shear viscosity coefficients for a completely ionized gas. We obtain their respective graphics considering an ionized gas of helium.

La description de l’évolution à long-terme des systèmes astrophysiques auto-gravitants tels que les disques stellaires, fait aujourd’hui l’objet d’un regain d’intérêt sous l’impulsion de deux développements récents. Cela repose tout d’abord sur le succès de la théorie Lambda-CDM pour décrire la formation des grandes structures et leurs interactions avec le milieu circum-galactique. En outre, de nouveaux développements théoriques permettent maintenant de décrire précisément l’amplification des perturbations extérieures ou internes, et leurs effets sur la structure orbitale d’un système sur les temps cosmiques. Ces progrès complémentaires nous permettent d’aborder la question lancinante des rôles respectifs de l’inné et de l’acquis sur les propriétés observées des systèmes auto-gravitants. Cette thèse est consacrée à la description de ces dynamiques séculaires, notamment lorsque l’auto-gravité joue un rôle important. Deux formalismes de diffusion, externe et interne, seront présentés en détail, et appliqués à différents problèmes astrophysiques. Dans un premier temps, nous étudierons les disques stellaires discrets infiniment fins, et retrouverons la formation d’étroites arêtes d’orbites résonantes en accord avec les observations et les simulations numériques, par le biais de la première mise en oeuvre de l’équation de Balescu-Lenard. Nous considérerons ensuite le mécanisme d’épaississement spontané des disques stellaires sous l’effet du bruit de Poisson. Enfin, nous illustrerons comment ces formalismes permettent également de décrire la dynamique des étoiles orbitant un trou noir supermassif dans les centres galactiques. / Understanding the long-term evolution of self-gravitating astrophysical systems, such as for example stellar discs, is now a subject of renewed interest, motivated by the combination of two factors. On the one hand, we now have at our disposal the well established Lambda-CDM model to describe the formation of structures and their interactions with the circum-galactic environment. On the other hand, recent theoretical works now provide a precise description of the amplification of external disturbances and discreteness noise, as well as their effects on a system’s orbital structure over cosmic time. These two complementary developments now allow us to address the pressing question of the respective roles of nature vs. nurture in the establishment of the observed properties of self-gravitating systems. The purpose of the present thesis is to describe such secular dynamics in contexts where self-gravity is deemed important. Two frameworks of diffusion, either external or internal, will be presented in detail, and applied to various astrophysical systems. This thesis will first investigate the secular evolution of discrete razor-thin stellar discs and recover the formation of narrow ridges of resonant orbits in agreement with observations and numerical simulations, thanks to the first implementation of the Balescu-Lenard equation. The spontaneous thickening of stellar discs as a result of Poisson shot noise will also be investigated. Finally, we will illustrate how the same formalisms allow us to describe the dynamics of stars orbiting a central super massive black hole in galactic centres.

Les mathématiques sont comprises en tant qu’ensemble d’idées abstraites, dans le sens où le monde réel – ou plutôt réalité – n’a pas à intervenir. Pourtant, certains faits mathématiques observables dans des données expérimentales ou simulées peuvent être contre-intuitifs. La thèse est divisée en deux parties : premièrement, on étudie mathématiquement les matrices du genre celles dont nous avons discutées en biologie et finance. En particulier, nous mettons en évidence le fait contre-intuitif suivant : pour ces matrices, le vecteur propre associé à la plus haute valeur propre est très proche de la somme de chacune des lignes de la matrice, colonne par colonne. Nous discutons aussi d’applications en théorie des graphes avec bon nombre de simulations numériques. Dans un second temps, nous attaquons le problème des contacts géniques : à partir d’une carte de contact génique, un vrai défi actuel est de retrouver la structure tridimensionnelle de l’ADN. Nous proposons diverses méthodes d’analyse matricielle de données, dont une met en évidence l’existence, dans le noyau, de zones disjointes où les interactions sont de différents types. Ces zones sont des compartiments nucléaires. Avec d’autres données biologiques, nous mettons en évidence la fonction biologique de chacun de ces compartiments. Les outils d’analyses sont ceux utilisés en finance pour analyser des matrices d’auto-corrélation, ou même des séries temporelles. / Mathematics are understood as a set of abstract ideas, in the measure of the real world – or reality – has no way to intervene. However, some observable mathematical facts in experimental or simulated data can be counter-intuitive. The PhD is divided into two parts: first, we mathematically study the matrices of the same type of the ones in biology and finance. In particular, we show the following counter-intuitive fact: for these matrices, the eigenvector associated with the highest eigenvalue is close to the sum of each row, column by column. We also discuss some applications to graph theory with many numerical simulations and data analysis.On the other hand, we will face the genetic contact problem: from a contact map, a real current challenge is to find the DNA 3D-structure. We propose several matrix analysis methods, which one show disjoinct areas in the nucleus where the DNA interactions are different. These areas are nuclear compartments. With other biological features, we characterize the biological function of each of the compartments. The analysis tools are the ones already used in finance to analyze the autocorrelation matrices, or even time series.

No description available.

This dissertation consists of three research projects of kinetic models: a structure preserving scheme for Poisson-Nernst-Planck equations and two efficient spectral methods for multi-dimensional Boltzmann equation.<br><br>The Poisson-Nernst-Planck (PNP) equations is widely used to describe the dynamics of ion transport in ion channels. We introduce a structure-preserving semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserving the steady-state). <br><br>Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. In this dissertation, we studied the state of the art in the fast Fourier method and discussed its limitation. Next, we proposed a new approach to implement the Fourier method, which can resolve those issues. <br><br>However, the Fourier method requires a domain truncation which is unphysical since the collision operator is defined in whole space R^d . In the last part of this dissertation, we introduce a Petrov-Galerkin spectral method for the Boltzmann equation in the unbounded domain. The basis functions (both test and trial functions) are carefully chosen mapped Chebyshev functions to obtain desired convergence and conservation properties. Furthermore, thanks to the close relationship of the Chebyshev functions and the Fourier cosine series, we can construct a fast algorithm with the help of the non-uniform fast Fourier transform (NUFFT).<br>

Orientador: Aguinaldo Robinson de Souza / Banca: Americo Sheitiro Tabata / Banca: Nelson Henrique Morgon / O Programa de Pós Graduação em Ciência e Tecnologia de Materiais, PosMat, tem caráter institucional e integra as atividades de pesquisa em materiais de diversos campi / Resumo: A teoria cinética dos gases tem como primeira aproximação o comportamento observado considerando o gás como tendo um comportamento ideal, isto é, pode ser modelado através da lei do gás ideal. As equações de estado conhecidas, como a equação do gás ideal e de van der Waals descrevem, dentro das aproximações do modelo, situações bastante diversas. Neste trabalho iremos utilizar a lei dos gases ideais, ou mais especificamente a implementação da Equação de van der Waals para o entendimento do fenômeno de colisões que entre átomos e moléculas / Abstract: The kinetic theory of gases has a a first approximation the observed behavior considering the gas to have an ideal behavior, ie it can be modeled by the ideal bas law. The equations of state known as the ideal gas equation and Van der Waals describe, within the approximations of the model, very different situations. In this paper we use the ideal gas law, or more specifically the implementation of the Vann der Waals equation for understandign the phenomenon of collisions between atoms and molecules / Mestre