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We obtain the characteristic relaxation rates and relaxation modes of a Bose gas in three regimes. The classical regime corresponds to a classical gas of hard spheres and the quantum regime corresponds to an interacting quantum Bose gas with no Bose-Einstein condensate present. In the condensed regime a Bose-Einstein condensate is present and modifies the behavior of the gas. In each regime there is a different kinetic equation that describes the evolution of the relevant distribution function. The classical kinetic equation is the Boltzmann equation and the quantum kinetic equation with no condensate present is the Uehling-Uhlenbeck equation. When a condensate is present, we derive a new kinetic equation that describes the evolution of the momentum distribution of Bogoliubov excitations or bogolons. For each of the three kinetic equations, we linearize the collision integral and use it to generate the elements of a collision matrix. The eigenvalues of this matrix give us the characteristic relaxation rates and the eigenvectors give us the relaxation modes. We report numerical results for the eigenvalues in each regime as the particle species, density and temperature of the gas are varied. / text

Collision strength, the measure of strength for a binary collision, hasn't been defined clearly. In practice, many physical arguments have been employed for the purpose and taken for granted. A scattering angle has been widely and intensively used as a measure of collision strength in plasma physics for years. The result of this is complication and unnecessary approximation in deriving some of the basic kinetic equations and in calculating some of the basic physical terms. The Boltzmann equation has a five-fold integral collision term that is complicated. Chandrasekhar and Spitzer's approaches to the linear Fokker-Planck coefficients have several approximations. An effective variable-change technique has been developed in this dissertation as an alternative to scattering angle as the measure of collision strength. By introducing the square of the reduced impulse or its equivalencies as a collision strength variable, many plasma calculations have been simplified. The five-fold linear Boltzmann collision integral and linearized Boltzmann collision integral are simplified to three-fold integrals. The arbitrary order linear Fokker-Planck coefficients are calculated and expressed in a uniform expression. The new theory provides a simple and exact method for describing the equilibrium plasma collision rate, and a precise calculation of the equilibrium relaxation time. It generalizes bimolecular collision reaction rate theory to a reaction rate theory for plasmas. A simple formula of high precision with wide temperature range has been developed for electron impact ionization rates for carbon atoms and ions. The universality of the concept of collision strength is emphasized. This dissertation will show how Arrhenius' chemical reaction rate theory and Thomson's ionization theory can be unified as one single theory under the concept of collision strength, and how many important physical terms in different disciplines, such as activation energy in chemical reaction theory, ionization energy in Thomson's ionization theory, and the Coulomb logarithm in plasma physics, can be unified into a single one -- the threshold value of collision strength. The collision strength, which is a measure of a transfer of momentum in units of energy, can be used to reconcile the differences between Descartes' opinion and Leibnitz's opinion about the "true'' measure of a force. Like Newton's second law, which provides an instantaneous measure of a force, collision strength, as a cumulative measure of a force, can be regarded as part of a law of force in general.

We investigate charmonium production in the hot medium created by heavy-ion collisions by setting up a framework in which in-medium charmonium properties are constrained by thermal lattice QCD (lQCD) and subsequently implemented into kinetic approaches. A Boltzmann transport equation is employed to describe the time evolution of the charmonium phase space distribution with the loss and gain term accounting for charmonium dissociation and regeneration (from charm quarks), respectively. The momentum dependence of the charmonium dissociation rate is worked out. The dominant process for in-medium charmonium regeneration is found to be a 3-to-2 process. Its corresponding regeneration rates from different input charmquark momentum spectra are evaluated. Experimental data on J/[psi] production at CERN-SPS and BNL-RHIC are compared with our numerical results in terms of both rapidity-dependent inclusive yields and transverse momentum (pt) spectra. Within current uncertainties from (interpreting) lQCD data and from input charm-quark spectra the centrality dependence of J/[psi] production at SPS and RHIC (for both mid-and forward rapidity) is reasonably well reproduced. The J/[psi] pt data are shown to have a discriminating power for in-medium charmonium properties as inferred from different interpretations of lQCD results.

<div><div>In this thesis, we focus on two challenges arising in kinetic equations, high dimensions and uncertainties. To reduce the dimensions, we proposed efficient methods for linear Boltzmann and full Boltzmann equations based on dynamic low-rank frameworks. For linear Boltzmann equation, we proposed a method that is based on macro-micro decomposition of the equation; the low-rank approximation is only used for the micro part of the solution. The time and spatial discretizations are done properly so that the overall scheme is second-order accurate (in both the fully kinetic and the limit regime) and asymptotic-preserving (AP). That is, in the diffusive regime, the scheme becomes a macroscopic solver for the limiting diffusion equation that automatically captures the low-rank structure of the solution. Moreover, the method can be implemented in a fully explicit way and is thus significantly more efficient compared to the previous state of the art. We demonstrate the accuracy and efficiency of the proposed low-rank method by a number of four-dimensional (two dimensions in physical space and two dimensions in velocity space) simulations. We further study the adaptivity of low-rank methods in full Boltzmann equation. We proposed a highly efficient adaptive low- rank method in Boltzmann equation for computations of steady state solutions. The main novelties of this approach are: On one hand, to the best of our knowledge, the dynamic low- rank integrator hasn’t been applied to full Boltzmann equation till date. The full collision operator is local in spatial variable while the convection part is local in velocity variable. This separated nature is well-suited for low-rank methods. Compared with full grid method (finite difference, finite volume,...), the dynamic low-rank method can avoid the full computations of collision operators in each spatial grid/elements. Resultingly, it can achieve much better efficiency especially for some low rank flows (e.g. normal shock wave). On the other hand, our adaptive low-rank method uses a novel dynamic thresholding strategy to adaptively control the computational rank to achieve better efficiency especially for steady state solutions. We demonstrate the accuracy and efficiency of the proposed adaptive low rank method by a number of 1D/2D Maxwell molecule benchmark tests. On the other hand, for kinetic equations with uncertainties, we focus on non-intrusive sampling methods where we are able to inherit good properties (AP, positivity preserving) from existing deterministic solvers. We propose a control variate multilevel Monte Carlo method for the kinetic BGK model of the Boltzmann equation subject to random inputs. The method combines a multilevel Monte Carlo technique with the computation of the optimal control variate multipliers derived from local or global variance minimization prob- lems. Consistency and convergence analysis for the method equipped with a second-order positivity-preserving and asymptotic-preserving scheme in space and time is also performed. Various numerical examples confirm that the optimized multilevel Monte Carlo method outperforms the classical multilevel Monte Carlo method especially for problems with dis- continuities<br></div></div>

La réduction fluide des équations cinétiques est un procédé couramment utilisé en physique des plasmas qui a pour objectif de remplacer la fonction de distribution définie dans l'espace des phases par des grandeurs fluides comme la densité et la pression. Cette réduction diminue la complexité du système initial. En contrepartie, la réduction fluide s'accompagne de la nécessité d'effectuer une fermeture sur les moments d'ordre supérieur. Celle-ci est souvent construite ad hoc en se basant sur des arguments physiques (e.g., quantités conservées, existance d'un théorème H, ...). Dans ce manuscrit, on propose un procédé de réduction qui permet de préserver la structure hamiltonienne du modèle cinétique parent. Ceci est important pour assurer qu'aucune dissipation d'origine non physique est introduite dans le modèle fluide, le munissant ainsi d'une structure hamiltonienne dont l'origine peut être suivie jusqu'à celle de la dynamique microscopique des particules. On utilise cette méthode pour construire des modèles fluides non-adiabatiques pour les trois premiers moments de la fonction de distribution associée à l'équation de Vlasov-Poisson à une dimension, i.e., la densité, la vitesse fluide et la pression. Les résultats sont ensuite étendus pour inclure la dynamique du flux de chaleur en considérant des fermetures construites à partir de l'analyse dimensionnelle. On montre également, pour un nombre arbitraire de champs, la relation existant avec le modèle water-bags. L'extension à des dimensions supérieures est étudiée dans le cadre de l'équation drift-cinétique ainsi que de l'équation de Vlasov-Poisson à trois dimensions. / Fluid reduction of kinetic equations is a ubiquitous procedure in plasma physics which aims to replace the distribution function defined in phase space with more concrete fluid quantities defined solely in configuration space such as the density, the fluid velocity and the pressure. This reduction lowers the complexity of the initial system, leading to a gain of physical insight into the phenomena under investigation as well as a significant decrease of the cost of numerical simulations. On the other hand, in order for the fluid reduction to be complete, one needs to perform a closure on the higher order fluid moments. The choice of the closure usually relies on some ad hoc physical arguments (e.g., conserved quantities, existence of an H-theorem, ...). In this manuscript, we present a reduction procedure that preserves the Hamiltonian structure of the parent kinetic model. This is important in order to ensure that no non-physical dissipation is introduced in the resulting fluid model, providing it with a geometric structure that can be traced back to the microscopic dynamics of the particles. We use this procedure to derive non-adiabatic fluid models for the first three fluid moments of the distribution function of the one dimensional Vlasov-Poisson equation, namely the density, the fluid velocity and the pressure. The results are extended to include the dynamics of the heat-flux by considering a closure based on dimensional analysis. For an arbitrary number of fields, we demonstrate the relationship with the water-bags model. Finally, the extension to higher dimensions is investigated through the drift-kinetic equation and the three dimensional Vlasov-Poisson equation.

Les traitements en radiothérapie consistent à irradier le patient avec desfaisceaux de particules énergétiques (typiquement des photons) ciblant la tumeur. Cesparticules sont transporté à travers le milieu et y dépose de l'énergie. Cette énergiedéposée, appelée la dose, est responsable des effets biologiques des radiations.Ce travail a pour but de développer des méthodes numériques de calcul etd'optimisation de la dose qui sont compétitives en termes de coût de calcul et de précisionpar rapport à des méthodes de référence.Le mouvement des particules est d'abord étudié via un système d'équationscinétiques linéaires. Cependant, résoudre directement ces systèmes est numériquementtrop coûteux pour des applications médicales. Pour palier ce coût de calcul, la méthodemoment, et en particulier les modèles Mn, est utilisée. Ces équations aux moments sontnon linéaires et valides sous une condition appelée réalisabilité.Les schémas numériques standards pour les équations aux moments sontcontraints par des conditions de stabilité qui se trouvent être très restrictives lorsque lemilieu contient des zones sous-denses. Des schémas numériques inconditionnellementstables et adaptés aux équations aux moments (préservant la réalisation) sontdéveloppés. Ces schémas se révèlent compétitifs en termes de coûts de calcul par rapportaux approches de référence. Finalement, ces méthodes sont appliquées dans uneprocédure d'optimisation visant à maximiser la dose dans la tumeur et la minimiser dansles tissus sains. / Radiotherapy treatments consists in irradiating the patient with beams ofenergetic particles (typically photons) targeting the tumor. Such particles are transportedthrough the medium and deposit energy in the medium. This deposited energy is the socalleddose, responsible for the biological effect of the radiations.The present work aim to develop numerical methods for dose computation andoptimization that are competitive in terms of computational cost and accuracy compared toreference method.The motion of particles is first studied through a system of linear transport equationsat the kinetic level. However, solving directly such systems is numerically too costly formedical application. Instead, the moment method is used with a special focus on the Mnmodels. Those moment equations are non-linear and valid under a condition calledrealizability.Standard numerical schemes for moment equations are constrained by stabilityconditions which happen to be very restrictive when the medium contains low densityregions. Inconditionally stable numerical schemes adapted to moment equations(preserving the realizability property) are developped. Those schemes are shown to becompetitive in terms of computational costs compared to reference approaches. Finallythey are applied to in an optimization procedure aiming to maximize the dose in the tumorand to minimize the dose in healthy tissues.

An attempt to numerically predict flow rate of experimental configuration of orifices in transition between molecular and viscous flow regime is described in detail. Discretization of Boltzmann kinetic equation known as lattice-Boltzmann method is derived and applied unfortunately with very little connection to the original experimental problem due to nearly supersonic nature of the experimental setup. Current quite unsatisfactory state of the art of compressible lattice-Boltzmann method is also presented.

<p><strong>Apstrakt je obrađen tehnologijama za optičko prepoznavanje teksta (OCR).</strong></p><p>U radu je spektrofotometrijski ispitivana brzina oksidacije L-askorbinske kiseline. Tok reakcije je praćen snimanjem UV apsorpcionih spektara termostatiranih rastvora L-askorbinske kiseline u puferima različitih vrednosti rN: acetatnom (4-6). fosfatnom (5-8). fosfatno-citratnom (6-8), boratnom (8-11). glicinskom (9-11) i univerzalnom (4-11) !zveden je jednostavan oblik kinetičke jednačine reakcije vi&scaron;eg i razlomljenog reda. Dobijena jednačina je univerzalna, jer obuhvata jednačine nultog, prvog, drugog, trećeg i razlomljenog reda izračunate su vrednosti reda reakcije i one variraju sa promenom vrednosti rN unutar jednog puferskog sistema. Veza između vrednosti konstanti brzine oksidacije L-askorbinske kiseline i vrednosti rN ispitivanih pufera je linearna do vrednosti rN=10.00. Vrednosti koeficijenta pravca prave su različite za ispitivane pufere. Vrsta anjona prisutna u puferu kao i prisusutvo jona metala u kori&scaron;ćenim hemikalijama utiču na vrednosti konstante brzine oksidacije L-askorbinske kiseline i one su različite u raznim puferima istih vrednosti rN. Vrednosti konstante brzine oskidacije L-askorbinske kiseline su največe u univerzalnom puferu pri svim vrednostima rN. Prisutna sme&scaron;a anjona manje utiče na oksidaciju L-askorbinske kiseline nego pojedinačni anjoni, verovatno zbog među- sobnih interakcija.</p> / <p><strong>Abstract was processed by technology for Optical character recognition (OCR).</strong></p><p>The paper deats with the spectrophotometric study of L-ascorbic acid oxidation rate. The course of reaction has been observed by recording the UV absorption spectra of thermostated solution of L-ascorbic acid in buffers with different рН values: acetate (4-6), phosphate (5-8), phosphate-citrate (6-8), borate (8-11), glycine (9-11) and universal (4-11). A simple form of kinetic eguation of higher and fractional order reaction has been observed. The obtained eguation is universal as it comprises the eguations of zero, first, second, third and fractional order. The reaction order values have been calculated and they vary with change of рН values within each buffer system. The relation between the values of L-ascorbic acid oxidation rate constants and рН value of the studied buffers is linear up to рН value = 10.00. The straight line stopes are different for investigated buffers. The anion type present in a buffer as wett as the presence of metal ions in applied chemicals affect the values of L-ascorbic acid oxidation rate constant and they are different in different buffers with the same рН values. The values of L-ascorbic acid oxidation rate constant are the highest in the universal buffer at all рН values. The present anion mixture affects the L-ascorbtc acid oxidation to a tesser degree than single anions, presumabty due to mutuat interactions.</p>

Dans cette thèse, nous étudions la généralisation à une infinité de particules d'un modèle hamiltonien décrivant les interactions entre une particule et son environnement. Le milieu est considéré comme une superposition continue de membranes vibrantes. Au bout d'un certain temps, tout se passe comme si la particule était soumise à une force de frottement linéaire. Les équations obtenus pour un grand nombre de particules sont proches des équations de Vlasov. Dans un premier chapitre, on montre d'abord l'existence et l'unicité des solutions puis on s'intéresse à certains régimes asymptotiques; en faisant tendre la vitesse des ondes dans le milieu vers l'infini et en redimensionnant les échelles, on obtient à la limite une équation de Vlasov, on montre que si l'on modifie en plus une fonction paramètrisant le système, on obtient l'équation de Vlasov-Poisson attractive. Dans un deuxième chapitre, on ajoute un terme de diffusion à l'équation. Cela correspond à prendre en compte une agitation brownienne et un frottement linéaire sur les particules. Le principal résultat de ce chapitre est la convergence de la distribution de particules vers une unique distribution stationnaire. On montre la limite de diffusion pour ce nouveau système en faisant tendre simultanément la vitesse de propagation vers l'infini. On obtient une équation plus simple pour la densité spatiale. Dans le chapitre 3, nous montrons la validité des équations déjà étudiées par une limite de champ moyen. Dans le dernier chapitre, on étudie l'asymptotique en temps long de l'équation décrivant l'évolution de la densité spatiale obtenue dans le chapitre 2, des résultats faibles de convergence sont obtenus / The goal of this PhD is to study a generalisation of a model describing the interaction between a single particle and its environment. We consider an infinite number of particles represented by their distribution function. The environment is modelled by a vibrating scalar field which exchanges energy with the particles. In the single particle case, after a large time, the particle behaves as if it were subjected to a linear friction force driven by the environment. The equations that we obtain for a large number of particles are close to the Vlasov equation. In the first chapter, we prove that our new system has a unique solution. We then care about some asymptotic issues; if the wave velocity in the medium goes to infinity, adapting the scaling of the interaction, we connect our system with the Vlasov equation. Changing also continuously a function that parametrizes the model, we also connect our model with the attractive Vlasov-Poisson equation. In the second chapter, we add a diffusive term in our equation. It means that we consider that the particles are subjected to a friction force and a Brownian motion. Our main result states that the distribution function converges to the unique equilibrium distribution of the system. We also establish the diffusive limit making the wave velocity go to infinity at the same time. We find a simpler equation satisfied by the spatial density. In chapter 3, we prove the validity of both equations studied in the two first chapters by a mean field limit. The last chapter is devoted to studying the large time asymptotic properties of the equation that we obtained on the spatial density in chapter 2. We prove some weak convergence results