21 research outputs found
Gyrokinetics from variational averaging: existence and error bounds
The gyrokinetic paradigm in the long wavelength regime is reviewed from the
perspective of variational averaging (VA). The VA-method represents a third
pillar for averaging kinetic equations with highly-oscillatory characteristics,
besides classical averaging or Chapman-Enskog expansions. VA operates on the
level of the Lagrangian function and preserves the Hamiltonian structure of the
characteristics at all orders. We discuss the methodology of VA in detail by
means of charged-particle motion in a strong magnetic field. The application of
VA to a broader class of highly-oscillatory problems can be envisioned. For the
charged particle, we prove the existence of a coordinate map in phase space
that leads to a gyrokinetic Lagrangian at any order of the expansion, for
general external fields. We compute this map up to third order, independent of
the electromagnetic gauge. Moreover, an error bound for the solution of the
derived gyrokinetic equation with respect to the solution of the Vlasov
equation is provided, allowing to estimate the quality of the VA-approximation
in this particular case.Comment: Keywords: averaging methods, Vlasov equation, Lagrangian mechanics,
motion of charged particles, magnetized plasma
Modeling and simulation of spin-polarized transport at the kinetic and diffusive level
L'objectif de cette thèse est de contribuer à la compréhension des phénomènes de mouvement de l'électron induits par le spin. Ces phénomènes aparaissent lorsqu'un électron se déplace à travers un environnement (partiellement) magnétique, de telle sorte que son moment magnétique (spin) peut interagir avec l'environnement. La nature quantique pure du spin nécessite des modèles de transport qui traitent des effets comme la cohérence quantique, l'intrication (corrélation) et la dissipation quantique. Sur le niveau méso- et macroscopique, il n'est pas encore clair dans quelles circonstances ces effets quantiques du spin peut transparaitre. Le but de ce travail est, d'une part, de dériver des nouveaux modèles de transport de spin à partir des principes de base et, d'autre part, de développer des algorithmes numériques qui permettent de trouver une solution de ces modèles. Cette thèse se compose de quatre parties. La première partie introductive contient un aperçu des concepts fondamentaux liés au transport polarisé en spin, tels que la magnéto-résistance géante (GMR), le couple de transfert de spin dans les multi-couches magnétiques et le caractère matriciel des équations de transport qui prennent en compte la cohérence de spin. L'accent est mis sur la modélisation du couple de transfert de spin, qui représente l'intersection de ces concepts. En particulier, nous considérons pour sa description le modèle diffusif de Zhang-Levy-Fert (ZLF) qui se compose de l'équation de Landau-Lifshitz et d'une équation de diffusion matricielle pour le spin. Un schéma de différences finies est développé pour résoudre numériquement ce système non-linéaire dans des structures multi-couches. Le modèle est testé par comparaison des résultats obtenus aux données expérimentales récentes. Les parties deux et trois forment le noyau thématique de cette thèse. Dans la deuxième partie nous proposons une équation de Boltzmann matricielle qui permet la description de la cohérence de spin sur le niveau cinétique. La nouveauté est un opérateur de collision dans lequel les taux de transition de la quantité de mouvement sont modélisés par une matrice 2x2 hermitienne; par conséquent, les libre parcours moyens des électrons spin-up et spin-down sont représentés par les valeurs propres de cette matrice de scattering. Après une dérivation formelle de l'équation de Vlasov matricielle à partir de l'équation de Wigner, l'équation cinétique qui suit est étudiée en ce qui concerne l'existence, l'unicité et la positivé d'une solution. En outre, le nouveau opérateur de collision est étudié rigoureusement et la limite de diffusion tc -> 0, correspondant à l'annulation de la moyenne de temps de scattering, est effectué. Les équations de drift-diffusion matricielle qui sont obtenues représentent une amélioration par rapport au modèle traité dans la première partie. Ce dernier est obtenu dans la limite ou la différence entre les deux valeurs propres de la matrice de scattering va disparaître. La troisième partie est consacrée à l'obtention de l'opérateur de collision matricielle introduit auparavant, à partir des principes quantiques. Pour cela, nous augmentons l'équation de von Neumann d'un système composite par un terme dissipatif qui fait tendre l'opérateur de densité totale vers l'approximation de Born. En vertu de la prémisse que la relaxation est le processus dominant, on obtient une hiérarchie d'équations non-Markoviennes. Celles-ci découlent d'une expansion de l'opérateur de densité en termes de tr, le temps de relaxation. Dans la limite de Born-Markov, tr -> 0, l'équation de Lindblad est récupérée. Elle a la même structure que l'opérateur de collision proposé dans la deuxième partie. Cependant, l'équation de Lindblad est encore une équation microscopique; donc la prochaine étape serait de procéder à la limite semi-classique du résultat obtenu.Dans la quatrième partie nous procédons à une étude numérique d'un modèle quantique-diffusif de spin qui décrit le transport dans un gaz d'électrons bidimensionnel avec un couplage spin-orbite de Rashba. Ce modèle suppose que les électrons sont dans un état d'équilibre quantique sous la forme d'un opérateur de Maxwell. Nous présentons deux discrétisations espace-temps du modèle couplé par l'équation de Poisson. Dans une première étape on applique une discrétisation en temps et on montre que les systèmes sont bien définis. Ceux-ci sont basés sur un formalisme fonctionnel pour traiter les relations non-locales entre les densités de spin. Nous utilisons ensuite des discrétisations espace-temps pour simuler la dynamique dans une géométrie typique d'un transistor. Les approximations différences finies sont du premier ordre en temps et du second ordre en espace. Les fonctionnelles discrètes sont minimisée à l'aide d'un algorithme du gradient conjugué et la méthode de Newton est appliquée afin de trouver les minima dans la direction désirée.The aim of this thesis is to contribute to the understanding of spin-induced phenomena in electron motion. These phenomena arise when electrons move through a (partially) magnetic environment, in such a way that its magnetic moment (spin) may interact with the surroundings. The pure quantum nature of the spin requires transport models that deal with effects like quantum coherence, entanglement (correlation) and quantum dissipation. On the meso- and macroscopic level it is not yet clear under which circumstances these quantum effects may transpire. The purpose of this work is, on the one hand, to derive novel spin transport models from basic principles and, on the other hand, to develop numerical algorithms that allow for a solution of these new and other existing model equations. The thesis consists of four parts. The first part has introductory character; it comprises an overview of fundamental spin-related concepts in electronic transport such as the giant-magneto-resistance (GMR) effect, the spin-transfer torque in metallic magnetic multilayers and the matrix-character of transport equations that take spin-coherent electron states into account. Special emphasis is placed on the modeling of the spin-transfer torque which represents the intersection of these concepts. In particular, we consider the diffusive Zhang-Levy-Fert (ZLF) model, an exchange-torque model that consists of the Landau-Lifshitz equation and a heuristic matrix spin-diffusion equation. A finite difference scheme based on Strang operator splitting is developed that enables a numerical, self-consistent solution of this non-linear system within multilayer structures. Finally, the model is tested by comparison of numerical results to recent experimental data. Parts two and three are the thematic core of this thesis. In part two we propose a matrix-Boltzmann equation that allows for the description of spin-coherent electron transport on a kinetic level. The novelty here is a linear collision operator in which the transition rates from momentum k to momentum k' are modeled by a 2x2 Hermitian matrix; hence the mean-free paths of spin-up and spin-down electrons are represented by the eigenvalues of this scattering matrix. After a formal derivation of the matrix-Vlasov equation as the semi-classical limit of the one-electron Wigner equation, the ensuing kinetic equation is studied with regard to existence, uniqueness and positive semi-definiteness of a solution. Furthermore, the new collision operator is investigated rigorously and the diffusion limit tc -> 0 of the mean scattering time is performed. The obtained matrix drift-diffusion equations are an improvement over the heuristic spin-diffusive model treated in part one. The latter is obtained in the limit of identical eigenvalues of the scattering matrix. Part three is dedicated to a first step towards the derivation of the matrix collision operator, introduced in part two, from first principles. For this, we augment the von Neumann equation of a composite quantum system by a dissipative term that relaxes the total state operator towards the Born approximation. Under the premise that the relaxation is the dominant process we obtain a hierarchy of non-Markovian master equations. The latter arises from an expansion of the total state operator in powers of the relaxation time tr. In the Born-Markov limit tr -> 0 the Lindblad master equation is recovered. It has the same structure as the collision operator proposed in part two heuristically . However, the Lindblad equation is still a microscopic equation; thus the next step would be to carry out the semi-classical limit of the result obtained. In part four we perform a numerical study of a quantum-diffusive, two-component spin model of the transport in a two-dimensional electron gas with Rashba spin-orbit coupling. This model assumes the electrons to be in a quantum equilibrium state in the form of a Maxwellian operator. We present two space-time discretizations of the model which also comprise the Poisson equation. In a first step pure time discretization is applied in order to prove the well-posedness of the two schemes, both of which are based on a functional formalism to treat the non-local relations between spin densities via the chemical potentials. We then use fully space-time discrete schemes to simulate the dynamics in a typical transistor geometry. Finite difference approximations applied in these schemes are first order in time and second order in space. The discrete functionals introduced are minimized with the help of a conjugate gradient-based algorithm in which the Newton method is applied to find the desired line minima
Canonical momentum based numerical schemes for hybrid plasma models with kinetic ions and massless electrons
We study the canonical momentum based discretizations of a hybrid model with
kinetic ions and mass-less electrons. Two equivalent formulations of the hybrid
model are presented, in which the vector potentials are in different gauges and
the distribution functions depend on canonical momentum (not velocity).
Particle-in-cell methods are used for the distribution functions, and the
vector potentials are discretized by the finite element methods in the
framework of finite element exterior calculus. Splitting methods are used for
the time discretizations. It is illustrated that the second formulation is
numerically superior and the schemes constructed based on the anti-symmetric
bracket proposed have better conservation properties, although the filters can
be used to improve the schemes of the first formulation.Comment: 24 pages, 8 figure
Closure of the Strongly Magnetized Electron Fluid Equations in the Adiabatic Regime
International audienc
Modélisation et simulation du transport spin-polarisé sur le niveau cinétique et diffusive
TOULOUSE3-BU Sciences (315552104) / SudocSudocFranceF
A Drift-Asymptotic scheme for a fluid description of plasmas in strong magnetic fields
International audienc
Asymptotic study of an anisotropic Fokker-Planck collision operator in a strong magnetic field
The present paper is concerned with the derivation, via asymptotic studies, of a reduced hybrid model describing the anisotropic fusion plasma dynamics in tokamaks. The parallel dynamics is governed by a kinetic equation, whereas the perpendicular dynamics is described by a Maxwellian distribution function, whose temperature T ⊥ satisfies an evolution equation, exchanging information with the parallel direction via some coupling terms. The reduced model is obtained from the underlying fully kinetic model, under the assumption of a strong magnetic field and strong collisionality in the perpendicular direction. From a numerical point of view, reduced models are very advantageous, permitting significant savings in computational times and memory. To improve the precision of the reduced description, we propose in this paper also first order correction terms with respect to the parameter describing the anisotropy, and discuss these terms from a physical point of view. This first order truncated model is new to our knowledge, meets the desired requirements of precision and efficiency, and its derivation is clearly exposed in this work, based on formal asymptotic studies
Asymptotic study of an anisotropic Fokker-Planck collision operator in a strong magnetic field
The present paper is concerned with the derivation, via asymptotic studies, of a reduced hybrid model describing the anisotropic fusion plasma dynamics in tokamaks. The parallel dynamics is governed by a kinetic equation, whereas the perpendicular dynamics is described by a Maxwellian distribution function, whose temperature T ⊥ satisfies an evolution equation, exchanging information with the parallel direction via some coupling terms. The reduced model is obtained from the underlying fully kinetic model, under the assumption of a strong magnetic field and strong collisionality in the perpendicular direction. From a numerical point of view, reduced models are very advantageous, permitting significant savings in computational times and memory. To improve the precision of the reduced description, we propose in this paper also first order correction terms with respect to the parameter describing the anisotropy, and discuss these terms from a physical point of view. This first order truncated model is new to our knowledge, meets the desired requirements of precision and efficiency, and its derivation is clearly exposed in this work, based on formal asymptotic studies