This thesis examines random variables of one dimension.

Certain numerical methods have been well developed for solving one-dimensional two-phase flow (e.g. gas and liquid) problems in the literatures during the last two decades. Based on the existing methods, the present work compares the computational efficiency, accuracy, and robustness of various numerical schemes by predicting the numerical solutions of fluid properties for a specific case to find the proper numerical method. One of the numerical schemes introduced in this work is a practical, semi-implicit upwind method used for fluid flow simulations in different flow patterns,stratified flow and slug flow. This method implements the iterative and non-iterative schemes using a two-fluid model that consists of sets of non-hyperbolic equations. A numerical error term is applied in the pressure equation to maintain the volume balance of the two-phase flow model. If the temperature varies, the discretised energy equations use similar error terms as in the pressure equation. In some cases, the small values of the numerical errors are negligible and do not influence the numerical results. These errors are, however, important factors to consider when maintaining the stability and robustness of the above numerical schemes for strong non-linear cases. The computational efficiency ofthe non-iterative scheme, where the inner iterations are deactivated, is better than the iterative scheme. Different grid arrangements are compared with respect to computational accuracy and efficiency. A staggered structured grid implements the same semi-implicit upwind method as in the non-iterative scheme; the non-staggered grid arrangement uses an existing flux-splitting scheme (Evje and Flåtten, 2003) as a reference. All the above schemes produce numerical solutions with a single precision that normally satisfy the requirements of computational accuracy of industrial two-phase pipe flows. However, if one pursues a higher-order accuracy scheme, e.g. a Roe-averaged algorithm, the governing equations should be strictly a hyperbolic system of partial differential equations, which is achieved by introducing the nonviscous force terms in the two-fluid model (LeVeque, 2002).By properly incorporating the non-conservative terms in the formulation of the numerical fluxes, the capability of the Roe-averaged algorithm is demonstrated by capturing shock waves. Results from the present research include the following. A one-dimensional scheme that solves a system of discretised equations with the staggered semi-implicit upwind method is presented and validated for its computational efficiencyand robustness. This scheme can be widely used in the industry with sufficient accuracy. The other first-order semi-implicit numerical schemes producestable numerical results, especially in the dynamic cases of two-phase flow, except when the gas phase nearly disappears or appears in pipes. The Roe-averaged algorithm is recommended due to the high-resolution numerical results obtained, but at the costs of computational time and effort.

No description available.

This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.

This thesis describes electrical measurements performed on low dimensional p-type devices, fabricated from GaAs/AlGaAs heterostructures. The Coulomb interaction between holes is similar to that between electrons. However, the kinetic energy is suppressed, which makes interaction effects particularly important. Holes may also be used to study band structure effects which arise from spin-orbit coupling in the valence band. The effects of Coulomb interactions in low dimensional electron systems are currently being studied extensively. Experiments presented in this thesis indicate the possible importance of Coulomb exchange interactions in both one and two dimensional hole systems (1DHSs,2DHSs). Tilted magnetic field studies of 2DHSs in the quantum Hall regime indicate that Landau levels at even filling factors will not cross. For high filling factor, this is attributed to a spin-orbit mixing effect which arises from the low symmetry ofthe system. At lower filling factor, activation-energy measurements verify that the energy gaps decrease and then increase as the field is tilted. However, the energy gap versus field dependences do not exhibit the curvature that might be expected from a perturbative anticrossing. It is speculated that the origin of this effect is a phase transition driven by the exchange interaction. Balanced arguments contrasting the relative strengths of the mixing and interactions theories are provided. The second part of this thesis describes a new method for the fabrication ofballistic 1DHSs, which exhibit clear conductance quantization. The quantization changes from even to odd multiples of e2/h as a function of the magnetic field in the plane of the heterostructure, as 'spin splitting' causes the 1D subbands to cross. Measurements of the 1D subband energy spacings are used together with the magnetic fields at which the crossings occur to calculate the in-plane g factors of the 1D subbands. These are found to increase as the number of occupied 1D subbands decreases. This enhancement of the g factor is attributed to exchange interactions; possible mixing explanations are also discussed. At higher magnetic fields, the pattern of quantization features shows that the subbands have crossed many times, and that the 1DHS can be strongly magnetized.

No description available.

This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.

In recent years, there has been an increasing interest in modeling crowd and evacuation dynamics. Pedestrian models are based on macroscopic or microscopic behavior. In this work, we are interested in developing models that can be used for evacuation control strategies. Hence, we use macroscopic modeling approach, where pedestrians are treated in an aggregate way and detailed interactions are overlooked. In this dissertation, we developed two-dimensional space crowd dynamic models to allow bi-directional low by modifying and enhancing various features of existing traffic and fluid dynamic models. In this work, four models based on continuum theory are developed, and conservation laws such as the continuity and momentum equations are used. The first model uses a single hyperbolic partial differential equation with a velocity-density relationship, while the other three models are systems of hyperbolic partial differential equations. For one of the system models presented, we show how it can be derived independently from a microscopic crowd model. The models are nonlinear, time-varying, hyperbolic partial differential equations, and the numerical simulation results given for the four macroscopic models were based on computational fluid dynamics schemes. We also started an initial control design that synthesizes the feedback linearization method for the one-dimensional traffic flow problem applied directly on the distributed parameter system. In addition, we suggest and discuss the information technology requirements for an evacuation system. This research was supported in part from the National Science Foundation through grant no. CMS-0428196 with Dr. S. C. Liu as the Program Director. This support is gratefully acknowledged. Any opinion, findings, and conclusions or recommendations expressed in this study are those of the writer and do not necessarily reflect the views of the National Science Foundation. / Ph. D.

Si les systèmes quantiques à une dimension ont longtemps été vus comme de simples modèle-jouets, bon nombre sont à présent réalisés dans les expériences d’atomes ultra-froids. Dans ces expériences, le potentiel de confinement du gaz induit nécessairement une inhomogénéité spatiale. Cette inhomogénéité brise l'invariance par translation qui joue un rôle clé dans les solutions analytiques, notamment celle de l'Ansatz de Bethe. On propose dans cette thèse de développer une théorie des champs effective à même de caractériser ces gaz quantiques inhomogènes, en généralisant la théorie du liquide de Luttinger. Dans ces conditions la métrique de l'action effective est courbe. Sous une hypothèse de séparation des échelles, les paramètres de l'action peuvent néanmoins être fixés par les solutions de l'Ansatz de Bethe. Le problème peut alors se ramener au cas d'un espace plat en faisant appel aux théories conformes. On est ainsi amené à résoudre le champ libre gaussien inhomogène, qui donne accès à toutes les fonctions de corrélations du modèle considéré. Dans cette thèse, on s'intéresse plus particulièrement au modèle de Lieb-Liniger. Les résultats obtenus sont comparés au système simulé par DMRG. / One-dimensional quantum systems have long been seen as simple toy-models but are nowadays often realized in ultracold atoms experiments. In those experiments the confining potential creates a spatial inhomogeneity. This breaks the translation invariance which plays a key role in exact analytical solutions as the Bethe Ansatz. In this thesis, we propose an effective theory generalizing the Luttinger liquid approach for inhomogeneous systems. In this setup, the effective action lives in curved space. However, making the hypothesis of separation of scales allow to compute the action's parameters by using Bethe Ansatz. The problem can then be solved in flat space by using tools from conformal theory. This leads us to solving the inhomogeneous gaussian free field that gives access to all correlation functions of the model under investigation. Here we focus on the Lieb-Liniger model. Our results are tested against DMRG simulations.

Ces dernières années d'importantes avancées techniques dans la manipulation des gaz atomiques ultrafroids ont ouvert la voie à la réalisation de fluides quantiques mésoscopiques de basse dimension. L'objet de cette thèse est l'étude théorique de certains systèmes mésoscopiques réalisables avec un gaz de Bose unidimensionel. Ces systèmes présentent des phénomènes quantiques intéressants, et sont potentiellement utiles en vue d'applications technologiques. Nous étudions le phénomène des courants permanents induits dans un gaz confiné sur un anneau par la rotation d'une barrière de potentiel, nous examinons la faisabilité d'un qubit fondé sur la superposition d'états de courant dans un réseau en forme d'anneau traversé par un champ de jauge et contenant un 'weak-link', ainsi que l'excitation dipolaire du gaz dans un 'split-trap' induit par le déplacement hors équilibre du potentiel externe. Dans tous ces cas, nous combinons diverses approches analytiques et numériques, qui permettent de couvrir l'ensemble des régimes d'interactions. Nous mettons en lumière un régime jusque-là inconnu, d'écrantage maximal des barrières de potentiel par le fluide, dû à une competition entre les effets des interactions et des fluctuations quantiques. Ces résultats ont des conséquences significatives sur le comportement de tels systèmes et, de ce fait, sont importants pour les réalisations en cours et à venir de dispositifs à gaz d'atomes ultrafroids. / Thanks to the experimental breakthrough of the last years in the manipulation of ultra cold atomic gases, it has become possible to realize low-dimensional and mesoscopic quantum fluids. The object of this thesis is the theoretical investigation of a few mesoscopic systems that can be realized with a one-dimensional Bose gas. These systems exhibit interesting quantum phenomena, and are potentially relevant for technological applications. We study the phenomenon of persistent currents induced by stirring the gas confined on a ring with a potential barrier, we examine the feasibility of a qubit based on the superposition of current states in a ring lattice threaded by a gauge field in the presence of a weak-link, and we investigate the dipole excitation of the gas in a split trap induced by an out-of-equilibrium displacement of the external potential. In all these cases, we apply a combination of analytical and numerical approaches that allow to cover all the interaction regimes. As a recurring theme, we disclose a so-far unknown regime of maximal screening of the barrier potential by the fluid, arising from the interplay of effects due to interactions and quantum fluctuations. These results have significant consequences for the behaviour of such systems and are important for the ongoing and future realization of ultracold atomic gases devices.