Étude des états fondamentaux du Laplacien magnétique avec annulation locale du champ

This thesis is devoted to the spectral analysis of the Schrödinger operator with magnetic field and semiclassical parameter, on a bounded regular domain $\Omega$ in dimension two, with Neumann boundary condition. We investigate the case when the magnetic field vanishes along a union of smooth curves. The aim is to understand the influence of the cancellation and to study the behaviour of the lowest eigenvalues and the associated eigenfunctions when $h$ tends to $0$. In this regime - called the semiclassical limit - the precise description of the eigenpairs requires the understanding of underlying models. In the first part, we consider a magnetic field which vanishes linearly along a smooth simple curve intersecting the boundary. The second part is devoted to the case when the magnetic field vanishes quadratically. In both cases, we firstly give a one term asymptotics of the lowest eigenvalue. The upper bound is obtained by using appropriate test functions whereas the lower bound results from a localisation process. This last aspect constitutes the most difficult part because of the different scales involved. Then we investigate the localisation properties of the first eigenfunctions thanks to semiclassical Agmon estimates. This leads to a full asymptotic expansion of the first eigenvalues. In the case when the magnetic field vanishes quadratically, we study in addition the model operator for which the cancellation set is a union of two straight lines, whose intersection form a non-zero angle. In the small angle regime, the structure of the spectrum is governed by an operator symbol with two parameters. We establish different properties of this symbol and the associated band function. Numerical simulations based on the finite elements library Mélina++ have guided the analysis and illustrate the obtained results. The difficulties of the numerical computations - induced by the high phase oscillations of the eigenfunctions - are circumvented by polynomial interpolation of high degree.Cette thèse concerne l'étude spectrale de l'opérateur de Schrödinger avec champ magnétique et paramètre semi-classique $h$, sur un domaine borné et régulier $\Omega$ en dimension $2$, avec condition de Neumann au bord. On s'intéresse plus particulièrement au cas où le champ magnétique s'annule sur une union de courbes régulières. L'objectif est de comprendre l'influence d'une annulation du champ et d'expliciter le comportement des basses valeurs propres et des fonctions propres associées lorsque $h$ tend vers $0$. Dans cette limite - dite semi-classique - la description précise des éléments propres passe par la compréhension de différents opérateurs modèles sous-jacents. La première partie est consacrée au cas d'un champ magnétique qui s'annule de manière non dégénérée le long d'une courbe régulière simple intersectant le bord du domaine. La deuxième partie concerne le cas d'une annulation quadratique à l'intérieur du domaine. Dans de ces deux cas d'étude, on donne dans un premier temps un équivalent asymptotique de la première valeur propre. La majoration s'obtient par une construction de fonctions tests appropriées tandis que la minoration s'obtient par une méthode de localisation quantique. Ce dernier aspect est délicat car il s'agit de gérer la transition entre des modèles ayant des homogénéités différentes. Dans un second temps, on examine les propriétés de localisation des premières fonctions propres, via des estimées d'Agmon semi-classiques. Ceci permet d'obtenir un développement asymptotique complet des premières valeurs propres, à n'importe quel ordre. Dans le cas d'une annulation quadratique, la thèse est complétée par une étude de l'opérateur modèle pour lequel le lieu d'annulation est une union de deux droites sécantes faisant un angle non nul. Dans la limite petit angle, la structure du spectre est gouvernée un symbole opérateur à deux paramètres. On établit différentes propriétés de ce symbole opérateur et de la fonction de bande associée. Des simulations numériques basées sur la librairie éléments finis Mélina++ ont guidé l'analyse et illustrent les résultats obtenus. Les difficultés numériques - dues aux fortes oscillations de la phase dans l'expression des fonctions propres - sont gérées grâce à une interpolation polynomiale de haut degré

On the semiclassical Laplacian with magnetic field having self-intersecting zero set

This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter h > 0 in the case when the magnetic field vanishes along a smooth curve which crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit h $\rightarrow$ 0. We show that each crossing point acts as a potential well, generating a new decay scale of h 3/2 for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems in R 2 for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to 0

Phase equilibrium properties of CO 2 /CH 4 mixed gas hydroquinone clathrates: Experimental data and model predictions

Hydroquinone (HQ) clathrates seem to be promising inclusion compounds for selective CO2 capture from gas mixtures. However, to date no phase equilibrium data are known in literature for mixed-gas HQ clathrates. This study presents experimental equilibrium pressures obtained within a range of 298–343 K for different CO2/CH4 gas mixtures. The clathrate composition is given for each equilibrium point. The capture selectivity is calculated from the molar composition of the CO2/CH4 gas mixture in the clathrate and in the gas phase. The results obtained reveal that CH4 molecules in the CO2/CH4 mixtures are preferentially captured at equilibrium conditions. Our experimental data are compared against numerical predictions obtained from thermodynamic modeling using the Conde’s model. Very good agreement is found between the calculated and experimental data in terms of clathrate phase equilibria

Molecular dynamics and thermodynamical modelling using SAFT-VR to predict hydrate phase equilibria : application to CO2 hydrates

This work was dedicated to the prediction of the three phase coexistence line (CO2 hydrate–liquid H2Oliquid/vapour CO2) for the H2O+CO2 binary mixture by using (i) molecular dynamics simulations, and (ii) the well known van der Waals-Platteeuw (vdWP) model combined with the SAFT-VR equation of state. Molecular dynamics simulations have been performed using the simulation package GROMACS. The temperature at which the three phases are in equilibrium was determined for different pressures, by using direct coexistence simulations. Carbon dioxide was modelled as a linear-rigid chain molecule with three chemical units, the well-known version TraPPE molecular model. The TIP4P/Ice model was used for water. To perform the thermodynamical modelling, the SAFT-VR EOS was incorporated in the vdWP framework. The values of the cell model parameters were regressed and discussed together with the influence of some assumptions of the vdWP model. Since SAFT-VR can describe most of fluids involved in hydrate modelling (inhibitors, salts…), this study is a first step in the description of hydrate forming conditions of more complex systems. Finally, the three-phase coexistence temperatures obtained with both simulations and theory at different pressures were compared with experimental result

CO2 Capture and Storage by Hydroquinone Clathrate Formation: Thermodynamic and Kinetic Studies

Hydroquinone (HQ) can form a gas clathrate in specific pressure and temperature conditions in the presence of CO2 molecules. This study presents experimental data of clathrate phase equilibrium and storage capacity for the CO2-HQ system in the range of temperature from about 288 to 354 K. Intercalation enthalpy and entropy are determined using the obtained equilibrium data and the Langmuir adsorption model. On a kinetic point of view, CO2-HQ clathrate formation by solid/gas reaction revealed a non-negligible effect of textural parameters on enclathration rate

Experimental Determination of Phase Equilibria and Occupancies for CO2, CH4, and N2 Hydroquinone Clathrates

Hydroquinone (HQ) forms organic clathrates in the presence of various gas molecules in specific thermodynamic conditions. For some systems, clathrate phase equilibrium and occupancy data are very scarce or inexistent in literature to date. This work presents experimental results obtained for the CO2–HQ, CH4–HQ, and N2–HQ clathrates, in an extended range of temperature from about 288 to 354 K. Formation/dissociation pressures, and occupancies at the equilibrium clathrate forming conditions, were determined for these systems. Experiments showing the influence of the crystallization solvent, and the effect of the gas pressure on HQ solubility, were also presented and discussed. A good agreement is obtained between our experimental results and the already published experimental and modeling data. Our results show a clear dependency of the clathrate occupancy with temperature. The equilibrium curves obtained for CO2–HQ and CH4–HQ clathrates were found to be very close to each other. The results presented in this study, obtained in a relatively large temperature range, are new and important to the field of organic clathrates with potential impact on gas separation, energy storage, and transport

Eigenstates of the Neumann magnetic Laplacian with vanishing magnetic field

Cette thèse concerne l'étude spectrale de l'opérateur de Schrödinger avec champ magnétique et paramètre semi-classique, sur un domaine borné et régulier en dimension 2, avec condition de Neumann au bord. On s'intéresse plus particulièrement au cas où le champ magnétique s'annule sur une union de courbes régulières. L'objectif est de comprendre l'influence d'une annulation du champ et d'expliciter le comportement des basses valeurs propres et des fonctions propres associées lorsque le paramètre semi-classique tend vers 0. Dans cette limite - dite semi-classique - la description précise des éléments propres passe par la compréhension de différents opérateurs modèles sous-jacents. La première partie est consacrée au cas d'un champ magnétique qui s'annule de manière non dégénérée le long d'une courbe régulière simple intersectant le bord du domaine. La deuxième partie concerne le cas d'une annulation quadratique à l'intérieur du domaine. Dans de ces deux cas d'étude, on donne dans un premier temps un équivalent asymptotique de la première valeur propre. La majoration s'obtient par une construction de fonctions tests appropriées tandis que la minoration s'obtient par une méthode de localisation quantique. Ce dernier aspect est délicat car il s'agit de gérer la transition entre des modèles ayant des homogénéités différentes. Dans un second temps, on examine les propriétés de localisation des premières fonctions propres, via des estimées d'Agmon semi-classiques. Ceci permet d'obtenir un développement asymptotique complet des premières valeurs propres, à n'importe quel ordre. Dans le cas d'une annulation quadratique, la thèse est complétée par une étude de l'opérateur modèle pour lequel le lieu d'annulation est une union de deux droites sécantes faisant un angle non nul. Dans la limite petit angle, la structure du spectre est gouvernée un symbole opérateur à deux paramètres. On établit différentes propriétés de ce symbole opérateur et de la fonction de bande associée. Des simulations numériques basées sur la librairie éléments finis Mélina++ ont guidé l'analyse et illustrent les résultats obtenus. Les difficultés numériques - dues aux fortes oscillations de la phase dans l'expression des fonctions propres - sont gérées grâce à une interpolation polynomiale de haut degré.This thesis is devoted to the spectral analysis of the Schrödinger operator with magnetic field and semiclassical parameter, on a bounded regular domain in dimension two, with Neumann boundary condition. We investigate the case when the magnetic field vanishes along a union of smooth curves. The aim is to understand the influence of the cancellation and to study the behaviour of the lowest eigenvalues and the associated eigenfunctions when the semiclassical parameter tends to 0. In this regime - called the semiclassical limit - the precise description of the eigenpairs requires the understanding of underlying models. In the first part, we consider a magnetic field which vanishes linearly along a smooth simple curve intersecting the boundary. The second part is devoted to the case when the magnetic field vanishes quadratically. In both cases, we firstly give a one term asymptotics of the lowest eigenvalue. The upper bound is obtained by using appropriate test functions whereas the lower bound results from a localisation process. This last aspect constitutes the most difficult part because of the different scales involved. Then we investigate the localisation properties of the first eigenfunctions thanks to semiclassical Agmon estimates. This leads to a full asymptotic expansion of the first eigenvalues. In the case when the magnetic field vanishes quadratically, we study in addition the model operator for which the cancellation set is a union of two straight lines, whose intersection form a non-zero angle. In the small angle regime, the structure of the spectrum is governed by an operator symbol with two parameters. We establish different properties of this symbol and the associated band function. Numerical simulations based on the finite elements library Mélina++ have guided the analysis and illustrate the obtained results. The difficulties of the numerical computations - induced by the high phase oscillations of the eigenfunctions - are circumvented by polynomial interpolation of high degree

On the semiclassical Laplacian with magnetic field having self-intersecting zero set

International audienceThis paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter $h > 0$ in the case when the magnetic field vanishes along a smooth curve which crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit $h \to 0$. We show that each crossing point acts as a potential well, generating a new decay scale of $h^{3/2}$ for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems in $\mathbb{R}^2$ for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to 0

Modélisation des équilibres de phases du clathrate d’hydroquinone.

Modélisation des équilibres de phases du clathrate d’hydroquinon