Academic literature on the topic 'Landau lowest level equation'

In the framework of collective field theory, We apply the Chern-Simon field theory treatment to the constraint equation for the lowest Landau level to investigate the generic properties for the quasi-particles of the FQH system. It shows a transparent connection to the Laughlin's wave functions. If we take an average over the wave functional for the constraint equation, the resulted equation can be interpreted as the vortex equation for the fractionally charged quasi-particles. Introducing a generalized ρ (density)-ϑ (phase) transformation, not only the fractional statistics and the hierarchy scheme can be drawn from the constraint equation, it also gives rise an interesting picture that vortices condense as a Halperin like wave fuction on a Laughlin like background condensate of ν=1/m.

By performing the Gupta–Bleuler quantization of a chiral boson, we obtain the chiral constraints, which correspond to the lowest Landau level conditions. From these, the chiral vacuum is defined as the vacuum of admixtures of many-harmonic oscillators. We construct the wave function for edge states of a droplet of incompressible quantum Hall fluid, by solving Schrödinger's equation on the basis of the chiral vacuum. This bosonic function can describe the collective edge modes, which are fundamentally a many-body effect of fermions at the lowest Landau level. In detail, the neutral edge state of FQHE is described by the α = 1 chiral boson theory. The charged edge states are described by the α ≠ 1 chiral boson theory.

We obtain expressions for the electric current in the Lowest Landau Level field theory in the presence of a general (external as well as inter-particle) interaction. This is done in the constrained Lagrangian formulation and is an extension of results obtained by Martinez and Stone for the external force case. However, we work directly with nonlocal field equations rather than convert the Lagrangian into a local one and use Noether's theorem.

Motivated by physics of the Lowest Landau Level and the Quantum Hall Effect, we investigate motion of an electron in a constant background magnetic field in the [Formula: see text]-Minkowski space-time. Starting from an action invariant under the noncommutative [Formula: see text] gauge transformations, we obtain the [Formula: see text]-deformed Dirac equation. Using the perturbative approach, we calculate noncommutative corrections to energy levels, mass and the gyromagnetic ratio up to the first order in the deformation parameter [Formula: see text].

We present an application of collective field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the no-free-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin’s wave function with and without quasi-hole excitations is also given.

Dyson equations for a Coulomb interacting anyon gas in a magnetic field are obtained and solved in the Hartree-Fock approximation. The states recently proposed by Greiter and Wilczek naturally arise from the Hartree-Fock solution. The electron self energies are calculated in the case when only the lowest Landau level is occupied.

Cette thèse a pour objet l'étude de l'équation du plus bas niveau de Landau, dans plusieurs contextes pertinents en physique et provient des modèles pour les condensats de Bose-Einstein. Nous nous penchons spécifiquement sur trois aspects liés à l'équation. Le premier est l'étude d'une classe de solutions appelées ondes stationnaires, à travers la minimisation d'une fonctionnelle énergie. Nous montrons notamment que la gaussienne est l'unique minimiseur global à symétries près pour un certain paramètre, à l'aide d'algèbre linéaire et bilinéaire. Le deuxième point concerne la conjecture de réseau d'Abrikosov. Nous investiguons l'équation avec ajout de conditions périodiques, et la linéarisons autour des réseaux. Nous aboutirons à la stabilité du réseau hexagonal. Le troisième et dernier aspect porte sur les ondes progressives pour l'équation couplée du plus bas niveau de Landau. Nous classifions de telles solutions ayant un nombre fini de zéros, et en déduisons l'existence de solutions dont les normes de Sobolev croissent
The aim of this thesis is to study the Lowest Landau Level equation, in several contexts relevant to physics and originating from models for Bose-Einstein condensates. In particular, we investigate three aspects of the equation. The first is the study of a class of solutions called stationary waves, through the minimization of an energy functional. In particular, we show that the Gaussian is the only global minimizer up to symmetries for a certain parameter, using linear and bilinear algebra tools. The second point concerns the Abrikosov lattice conjecture. We investigate the equation with the addition of periodic conditions, and linearize it around lattices. This results in the stability of the hexagonal lattice. The third and final aspect concerns progressive waves for the coupled Lowest Landau Level equation. We classify such solutions with a finite number of zeros, and deduce the existence of solutions with growing Sobolev norms

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

We have performed a theoretical search for the incompressible fractional quantum Hall e ect (FQHE) states in the partially lled lowest Landau level (LL), where the lling factor lies within the range 2=7 < < 2=5. Incompressibility has been proved by calculating energy spectrum at fully spin polarized 3/8, 3/10, 4/11, 4/13, 5/13 and 5/17 llings and at partially spin polarized 3/8 lling of the lowest LL. After nding the appropriate incompressible states, we have calculated their collective modes of excitation in a single mode approximation. This thesis is consisting of six chapters, organized as follows.
The research was carried out under supervision of Prof. S S Mondal of the Theoretical Physics division under SPS [School of Physical Sciences]
The research was conducted under CSIR grant and fellowship

The kinetic equation with the nonlocal shifts is presented as the final result on the way to derive the kinetic equation with nonlocal corrections. The exclusive dependence of the nonlocal and non-instant corrections on the scattering phase shift confirms the results from the theory of gases. With the approximation on the level of the Brueckner reaction matrix, the corresponding non-instant and nonlocal scattering integral in parallel with the classical Enskog’s equation, can be treated with Monte-Carlo simulation techniques. Neglecting the shifts, the Landau theory of quasiparticle transport appears. In this sense the presented kinetic theory unifies both approaches. An intrinsic symmetry is found from the optical theorem which allows for representing the collision integral equivalently either in particle-hole symmetric or space-time symmetric form.

The energies, kets and wave functions obtained from the Schrödinger equation for the hydrogen atom are examined in Chapter 9. Three quantum numbers are identified. The energies turn out to be the same as in the Bohr model, and an energy-level diagram appropriate to the quantum description is constructed. Graphs of the probability distributions are interpreted as the electron being in a “cloud” around the proton, rather than at a fixed position: the atom is fuzzy, not sharp-edged. The wavelengths of the five photons of the Balmer series are shown to be in the visible range. These photons are emitted when electrons transition from higher-excited states to the second lowest one, which means that electronic-type transitions underlie the presence of colors in our visible environment. The non-collapse of the atom, required by classical physics, is shown to arise from the structure of Schrödinger’s equation.

In the near term, biomass in various forms is the most available renewable energy source in the USA, particularly in the wild land-urban interface (WUI). With current high natural gas (NG) prices the possibility of gasifying biomass and developing a local Biomass Alliance with Natural Gas (BANG) so that the biomass gas (BG) can supplement natural gas (NG) could have many energy, environmental and economic (EEE) benefits. An analytic cost estimation (ACE) method is used to assess various BANG technologies potentially applicable at the WUI and to determine NG prices at which BG capable systems can deliver electricity at competitive costs. ACE is based upon the approximate linear relationship between cost of electricity (COE = Y), and cost of fuel (COF = X), i.e., Y = K + SX, as seen in many detailed cost analyses of electrical generating systems. ACE is here used to guide efforts directed towards energy sustainability in the WUI where nearby biomass stores are abundant. Thermal conversion and the use of the fuel to supplement NG is considered here at various power (P) levels to lower the cost of electricity. A reasonable K(P) and S(P) for NG fired electrical generation systems is first established. Then using an Antares Group Inc. report (AGIR) analyses of eleven biomass fueled technologies K(P) and S(P) are identified using only three adjusted parameters for each technology. An accurate analytical equation is also found for AGIR calculations of net present value (NPV = Z) vs. P and Y. These equations are then used to interpolate and extrapolate the AGIR economic analyses to other Ps and X or Y in some cases leading to other conclusions than in the AGIR. We conclude that BANG can save energy costs in many communities at the WUI and lead the way to: the development of economically competitive woody biomass supply and applications industries provide jobs, stabilize local economies, reduce USA’s dependence on imported fuels and lower greenhouse gas emissions.

Multilayer insulation (MLI) has the lowest thermal conductivity of any currently used insulation in high vacuum environments and is used in cryogenic insulation system to minimize heat leaks in liquid hydrogen storage tanks. MLI consists of highly reflective radiation shields separated by spacers or insulation. The thermal conductivity of MLI varies with both temperature and vacuum level. Most published apparent thermal conductivities were measured for temperatures between 80K and 300K; some of the published data were for temperatures between 20K and 80K. Since the temperature of liquid hydrogen is 20K and the storage tanks are exposed to ambient air, it is essential to know the thermal performance of MLI for the temperature range of 20K to 300K. In addition, in order to provide a detailed temperature distribution and to optimize insulation systems with respect to the number of layers of MLI, layer density, insulation weight, and separator configuration, the layer-by-layer thermal performance of MLI has to be established for efficient storage tank design. A general equation for thermal conductivity was developed based on heat transfer principles for a wide range of temperature differences and vacuum levels. The equation consists of four heat transfer modes: 1) thermal radiation between two adjacent reflectors, 2) thermal radiation absorbed by spacers 3) gas conduction, and 4) solid spacer conduction. The equation can be applied for the temperature ranges of liquid hydrogen up to ambient, and for pressure ranges between 1.33 mPa to 1.33 kPa (0.01 millitorr and 10 torr). The predicted layer-by-layer temperatures, heat fluxes and apparent thermal conductivities using the developed thermal conductivity equation show very good agreement with measured data between the temperatures of 80K and 300K at the various pressure levels. When the equation was applied for a temperature of 20K, heat fluxes increased due to the larger temperature difference, while apparent thermal conductivities decreased due to the lower cold side temperature.

An experimental investigation and a numerical simulation of aerodynamic-aeroacoustic performance of an axial-flow fan at the rotating stall condition are reported in this paper. The wake profiles of rotors were measured using two-channel, hot wire probes. The Reynolds-averaged Navier-Stokes equations of the flow field were numerically solved. The two rotors studied included a conventional and a skewed-swept rotor with the objective being to determine the influence of skew and sweep on the sound levels and rotating stall characteristic. In both cases, a single stall cell was observed in fully developed stall conditions. The experimental results showed a significant difference in stall cell hysteresis, size, circumference and spanwise position, and cell propagation velocity between the two rotors. The stall cell in the skewed rotor propagated faster than the one in the conventional rotor. A fully developed stall cell was observed near the mid-span in the skewed rotor whereas it was situated near the hub in the conventional rotor. The noise level in the skewed rotor at the stall condition was more than 2 dBA lower than in the conventional rotor. Sound frequency spectra were obtained and analyzed in the near and far fields between the skewed and conventional radial rotors at rotating stall point and at design point. Significant differences in the sound directivity between the two types of rotors at steady and unsteady operating conditions were observed throughout the measured sound region. Results showed that the numerical models predicted at the rotating stall condition with good accuracy.

Brush seals are considered as a category of compliant seals, which tolerate a great high level of interference between the seal and the rotor or shaft. Their superior leakage characteristics have opened many application fields in the turbo-machinery world, ranging from industrial steam turbines to jet engines. However, brush seal designers have to find a trade-off between the lower parasitic leakage but higher heat generation properties of brush seals for given operation conditions. As brush seals can maintain contact with the rotor for a wide range of operating conditions, the contact force/pressure generated at the seal-rotor interface becomes an important design parameter for sustained seal performance and longevity of its service life. Furthermore, due to this contact force at the interface, frictional heat generation is inevitable and must be evaluated for various design and operating conditions. In this paper, frictional heat generation at the sealrotor interface is studied. To capture temperature rise at the interface, a thermal image of the seal and rotor is taken with an infrared camera under various operating conditions. The temperature map of the rotor is compared to results from thermal finite element analysis of the rotor to back calculate the heat flux to the rotor. A closed form equation for frictional heat generation is suggested as a function of seal design parameters, material properties, friction coefficient and empirical factors from testing.

Capturing a level of modeling of the flow inside a multi-stage turbomachine, such as unsteadiness for example, can be done at different degrees of details, either by capturing all deterministic features of the flow with a pure unsteady method or by settling for an approximated solution at a lower computational cost. The harmonic methods stand in this second category. Amongst them the “Nonlinear Harmonic Method” from He revealed the most efficient. This method consists of solving the fully nonlinear 3D steady problem and a linearized perturbation system in the frequency domain. As it has been shown by the authors that the circumferential variations exhibit a harmonic behavior, it is proposed here to adapt this method to the through-flow model, where the main nonlinear system would be the common throughflow equations and the auxiliary system would give access to the circumferential stresses. As the numerical local explicit impermeability conditions are unsupported by Fourier series, the adaptation of this technique to the throughflow model passes through a reformulation of the blade effect by a smooth force field as in the “Immersed Boundary Method” from Peskin. A simple example of an inviscid flow around a cylinder will illustrate the preceding developments, bringing back the mean effect of the circumferential non uniformities into the meridional flow.

Abstract Creating product structure models is an important task in design, as different such models are needed during the product design process. Depending on what theoretical model or methodology for product design that is used during the process, structure models of different kinds are developed. In structural axiomatic design, functional requirement structure models and physical solution structure models are essential. These are hierarchial tree structure models which are developed during the process by zigzagging between the functional domain and the physical solution domain from the highest to the lowest hierarchical structure level. While doing this, functional couplings, making it harder to fulfill a certain functional requirement without affecting others, are built into the structure. The theory of axiomatic design provides tools to identify such couplings. In this work, the so called design equation in the axiomatic design theory, has been given a graphical interpretation. This has made it possible to use a computer based structure modeling tool, with a graphic user interface, to directly identify and analyze functional couplings in the structure model on the graphical screen.

We study the non-equilibrium dynamics of a pair of qubits made of two-level atoms separated in space with distance r and interacting with one common electromagnetic field but not directly with each other. Our calculation makes a weak coupling assumption, but no Born or Markov approximation. We derived a non-Markovian master equation for the evolution of the reduced density matrix of the two-qubit system after integrating out the electromagnetic field modes. It contains a Markovian part with a Lindblad type operator and a nonMarkovian contribution, the physics of which is the main focus of this study. We use the concurrence function as a measure of quantum entanglement between the two qubits. Two classes of states are studied in detail: Class A is a one parameter family of states which are the superposition of the highest energy |I〉 ≡ |11〉 and lowest energy |O〉 ≡ |00〉 states, υiz, |A〉≡p|I〉+(1−p)|O〉, with 0 ≤ p ≤ 1; and Class B states |B〉 are linear combinations of the symmetric |+〉=12(|01〉+|10〉) and the antisymmetric |−〉=12(|01〉−|10〉) Bell states.