Dans cette thèse, la connexion entre certaines structures algébriques récentes (algèbres tridiagonales, algèbre q-Onsager, algèbres q-Onsager généralisées), la théorie des représentations (paire tridiagonale, paire de Leonard, polynômes orthogonaux), certaines des propriétés de ces algèbres et l’analyse de modèles intégrables quantiques sur le réseau (la chaîne de spin XXZ ouverte aux racines de l’unité) est considérée. / In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, q-Onsager algebra, generalized q-Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal polynomials), some properties of these algebras and the analysis of related quantum integrable models on the lattice (the XXZ open spin chain at roots of unity) is considered.

The Onsager heat of transport for p-tert-butyltoluene was measured, as part of a series of preliminary experiments towards the determination of the importance of temperature gradients on the air-sea flux of carbon dioxide. The results presented in this thesis imply that the temperature gradient is a major contributor to the magnitude of the air-sea flux. The heat of transport has been measured for the p-tert-butyltoluene system by measuring stationary-state pressure changes for known temperature differences on the vapour side of the interface. At the pressure ranges used the number of mean free paths was always outside the Knudsen zone, but the values of Q* were approximately 100 % of the latent heat of vaporisation. Departures from linearity of plots of P against ΔT are attributed to temperature jumps at the surface of the dry upper plate. Both the results taken for p-tert-butyltoluene and the earlier results for water from this laboratory fit to a Type III BET isotherm, where the c parameter is not constant. They also reveal the importance of the temperature gradient in determining the value of the thermal accommodation coefficient, and provide a new method of measuring thermal accommodation coefficients for a variety of surfaces and vapours

The Onsager heat of transport for p-tert-butyltoluene was measured, as part of a series of preliminary experiments towards the determination of the importance of temperature gradients on the air-sea flux of carbon dioxide. The results presented in this thesis imply that the temperature gradient is a major contributor to the magnitude of the air-sea flux. The heat of transport has been measured for the p-tert-butyltoluene system by measuring stationary-state pressure changes for known temperature differences on the vapour side of the interface. At the pressure ranges used the number of mean free paths was always outside the Knudsen zone, but the values of Q* were approximately 100 % of the latent heat of vaporisation. Departures from linearity of plots of P against ΔT are attributed to temperature jumps at the surface of the dry upper plate. Both the results taken for p-tert-butyltoluene and the earlier results for water from this laboratory fit to a Type III BET isotherm, where the c parameter is not constant. They also reveal the importance of the temperature gradient in determining the value of the thermal accommodation coefficient, and provide a new method of measuring thermal accommodation coefficients for a variety of surfaces and vapours

En turbulence pleinement développée et incompressible, on constate que l’énergie cinétique d’unécoulement est dissipée à un taux indépendant du nombre de Reynolds. C’est la loi zéro de la turbulence.Cette loi, qui fut découverte en 1935 par Taylor, a eu de nombreuses confirmations expérimentaleset numériques, et est au coeur de notre compréhension de la physique des régimes turbulents. Dansles années qui suivirent, Taylor proposa un mécanisme pour rendre compte de la loi zéro, basé sur laviscosité et sur l’idée d’une cascade d’énergie à travers les échelles. En 1949, Onsager se rend comptequ’une dissipation d’énergie peut aussi se produire sans l’assistance des forces visqueuses à petite échellesi le champ de vitesse devient suffisamment irrégulier, et propose une conjecture sur la régularité minimaleque devrait satisfaire le champ de vitesse pour assurer la conversation de l’énergie en l’absencede viscosité. En 2000, deux mathématiciens français, Jean Duchon et Raoul Robert, formalise pour lapremière fois les idées d’Onsager dans un cadre mathématique rigoureux. Ils établissent la forme exactede la dissipation d’énergie émanant de l’existence possible de singularités, et I’expriment en fonctiondes incréments de vitesse. Cependant, la pertinence de ces concepts en turbulence expérimentale resteà établir, et n’a jamais été étudiée.Dans cette thèse, nous proposons les premiers tests des idées d’Onsager à partir de données expérimentales,en se basant sur le travail de Duchon et Robert. Pour cela, nous nous plaçons dans le cadredes écoulements de von Kármán où la régularité des équations de Navier-Stokes n’est pas connue. Nousutilisons des mesures de vélocimétrie par image de particules pour obtenir les trois composantes duchamp de vitesse dans un plan méridien, et ainsi calculer ses incréments à l’échelle de résolution de notresystème de mesure. Le résultat principal de ce travail est la mise en évidence du caractère non-trivialdes écoulements turbulents à l’échelle de Kolmogorov, où l’on observe des topologies très irrégulièresdu champ de vitesse coïncidant avec des évènements extrêmes de transferts inertiels d’énergie. / The zeroth law of turbulence states that fully developed turbulent incompressible flows dissipatetheir kinetic energy independently of the Reynolds number. Since its discovery by Taylor in 1935, thislaw has had many experimental and numerical confirmations, and is at the heart of our understandingof turbulence. In the following years, Taylor proposed a mechanism for the zeroth law, based onviscosity and the idea of a cascade of energy through scales. In 1949, Onsager realized that energydissipation could occur without the final assistance by viscosity at small scales if the velocity fieldbecomes sufficiently irregular, and conjectured the minimum regularity condition above which energyconservation is ensured in the absence of viscosity. In 2000, two french mathematicians, Jean Duchonand Raoul Robert, were able to derive the analytical expression for the inertial dissipation in termsof velocity increments, along with the corresponding energy balance. However, the relevance of theseideas for real turbulence has never been studied.In this thesis, we present the first tests of Onsager’s idea from experimental data, based on thework of Duchon and Robert. We enter the framework of von Kármán flows for which the regularity ofNavier-Stokes equations is unknown. We use particle image velocimetry measurements which provideus with the three components of the velocity field on a meridional plane, and allows for the computationof velocity increments at the resolution scale of our measurement set-up. In this work, we point out thenon-trivial character of turbulent flows at the Kolmogorov scale, where we observe irregular

In der vorliegenden Arbeit wurden drei unabhängige Problemfelder moderner biophysikalischer und materialwissenschaftlicher Forschung untersucht: Diffusion in binären Legierungen, der Umbauprozess in trabekulärem Knochen und die Voraussage mechanischer Eigenschaften, insbesonders der Biegesteifigkeit, selbstorganisierender, amphiphiler Membrane. Für alle drei Problemfelder wurden Gittermodelle gewählt, um ausgesuchte Fragestellungen zu untersuchen. Für den Fall der Diffusion in Legierungen war dies, inwieweit sich der Diffusionprozess, der sich auf atomarer Ebene als diskrete Platztäusche zwischen Atomen und Leerstellen manifestiert, auf einer größeren, makroskopischen, Ebene mit Hilfe einer kontinuumstheoretischen Theorie beschreiben lässt. Im Fall der Beschreibung des Umbauprozesses in in trabekulärem Knochen wurde die die spongiöse Architektur des Knochens auf ein Gitter abgebildet und mittels einer vereinfachten mechanischen Beschreibung die lokale Belastung in jedem Knochenelement bestimmt. Die zeitliche Entwicklung des Systems wurde mittels eines stochastischen Umbaugesetzes gesteuert, das die Wahrscheinlichkeit für Knochenan- bzw. -abbau als Funktion der lokalen Volumenänderung vorgab. Es wurde gezeigt, dass ein nicht-lineares Umbaugesetz bessere Übereinstimmung mit experimentellen Ergebnissen zeigt, als ein rein lineares. Weiters wurde das Krankheitsbild der Osteoporose untersucht und es konnte eine Unterscheidung zwischen einem normalen Alterungsprozess der Knochenstruktur und einer krankhaften Veränderung gezogen werden. Um die mechanischen Eigenschaften selbstorganisierender Membrane zu bestimmen, wurden linear elastische Federkräfte zwischen benachbarten Molekülen angenommen. Die volle elastische Matrix und daraus die gewünschten Eigenschaften wurden für unterschiedliche Zusammensetzungen der Membrane bestimmt. Es wurde gezeigt, dass die Biegesteifigkeit solcher Membrane in einem begrenzten Konzentrationsbereich um mehrere Größenordnungen variieren kann. / This thesis presents the results of investigations on three independent research topics of modern biophysical and materials science research: substitutional diffusion in binary alloys, the remodelling process in trabecular bone and the prediction of mechanical properties of self assembling, amphiphilic bilayers. The basic description of all three projects is based on lattice models, a highly successful class of models that are used in several fields of modern physics to describe physical processes. For the diffusional process in alloys, which on a microscopic scale manifests in a discrete site exchange between one atom and a neighbouring vacancy, it was investigated how this microscopic description can be reconciled with a macroscopic continuum model. For the investigations on remodelling of trabecular bone, the architecture of bone was mapped onto a lattice and the local mechanical state of each element was determined by a simplified mechanical model. A stochastic description was chosen to model the time evolution of the system, relating the probability of bone formation and resorption, respectively, to the local volume changes of the bone elements. It was shown that a non-linear remodelling law is a better candidate to describe the remodelling process in real bone than a linear one. Furthermore, applying the model to osteoporosis - a wide spread bone disease - it was shown that in the features attributed to osteoporosis one has to distinguish between normal ageing of bone''s architecture and additional changes that stem from pathological alterations in the regulatory system. A simple concept was introduced to model the mechanical properties of self-assembled membranes. The molecules forming the membrane are assumed to occupy a triangular lattice, nearest neighbours are connected by linear elastic springs. It was shown that the bending rigidity exhibits a pronounced concentration dependence, varying over orders of magnitude in a narrow concentration regime.

Este trabalho apresenta uma extensiva série de resultados experimentais da condutividade induzida por radiação X contínua, durante e após a irradiação, em amostras de PET e Kapton. As medidas foram realizadas variando-se uma série de parâmetros, tais como: o campo elétrico aplicado, a taxa de exposição, a espessura da amostra, o tipo de eletrodo, a energia da radiação e as condições ambientes. Foram feitas ainda medidas da corrente termo-estimulada em amostras irradiadas e não irradiadas, que permitiram verificar a presença de armadilhas nos materiais. Medidas da corrente fotônica com diferentes eletrodos e espessuras das amostras, constataram a influência do eletrodo no valor dessa corrente. Finalmente um modelo teórico foi desenvolvido, baseado na teoria de balanço dos portadores generalizada, com a inclusão do efeito do campo elétrico na taxa de geração de portadores (efeito Onsager). O ajuste teórico-experimental permitiu a determinação numérica dos principais parâmetros de condução, tais como, mobilidade dos portadores, coeficiente de recombinação e densidade de armadilhas, para os dois materiais estudados. / In this work we present extensive results of measurements of the prompt and delayed radiation-induced conductivity of samples of PET and Kapton. Experimental parameters, such as the effective energy of the radiation, the exposure rate, the total dose, the value of the applied electric field, the nature of the electrodes, and the ambienta1 conditions were changed within wide limits. We also report measurement of thermally stimulated currents for non-irradiated and for irradiated samples which allowed us to investigate the trap-structure of the materials. Measurements of photo-Compton currents with different electrode materials and sample thicknesses gave information about the relation between the nature of the electrodes and the amplitudes of the currents. Based on the generalized rate theory of radiation-induced conduction we developed a theoretical model which includes the effect of the applied electric field on the carrier generation yield (geminate recombination, Onsager effect). Comparison of experimental and theoretical curves allowed us to determine the values of the main conduction parameters, such as carrier mobility, recombination coefficient, trap densities, for the materials under investigation.

Neste trabalho estudamos, no contexto de campo médio, modelos microscópicos que possam descrever o comportamento termodinâmico das fases nemáticas em sistemas líquido-cristalinos. Considerando apenas interações atrativas, investigamos modelos de interações quadrupolares para objetos intrinsecamente biaxiais. Esses modelos apresentam mesofases nemáticas uniaxiais e biaxiais, pontos triplos e multicríticos (tricríticos, pontos de Landau, etc.). Ainda no contexto de forças atrativas, introduzimos um modelo de mistura binária de objetos intrinsecamente uniaxiais e objetos intrinsecamente biaxiais, numa formulação annealed. Essa mistura apresenta diagramas de fases bastante ricos, com topologias diversas, onde identificamos estruturas uniaxiais e biaxiais, fases reentrantes e uma grande variedade de pontos multicríticos (tricríticos, pontos críticos terminais, etc.). No caso de interações estéricas, estudamos uma teoria do funcional densidade para sistemas anisotrópicos densos, construída a partir de uma aproximação de van der Waals. Para esferoides prolatos, o modelo prevê um espaço de orientações com regiões não-acessíveis para as partículas. Além disso, o sistema apresenta uma região de coexistência entre as fases nemática e isotrópica. / We study, in a mean-field approximation, microscopic models which can lead to nematic liquid-crystalline phases. Considering attractive forces, we investigate models with quadrupolar interactions for intrinsically biaxial objects. These models present uniaxial and biaxial nematic mesophases, triple and multicritical points (tricritical point, Landau point, etc.). We also introduce a model for a binary mixture of intrinsically uniaxiail and biaxial objects, in an annealed treatment. The mixture exhibits phase diagrams with very rich topologies, where we find uniaxial and biaxial structures, reentrant phases and many different multicritical behaviors (tricritical point, critical endpoint, etc.). Moreover, assuming steric interactions, we investigate a density functional theory for hard anisotropic bodies at high densities, based on a van der Waals approximation. For hard spheroids, the model leads to an orientation space with forbidden regions for the particles. Also, the system phase separates in a nematic and an isotropic phases.

Neste trabalho estudamos, no contexto de campo médio, modelos microscópicos que possam descrever o comportamento termodinâmico das fases nemáticas em sistemas líquido-cristalinos. Considerando apenas interações atrativas, investigamos modelos de interações quadrupolares para objetos intrinsecamente biaxiais. Esses modelos apresentam mesofases nemáticas uniaxiais e biaxiais, pontos triplos e multicríticos (tricríticos, pontos de Landau, etc.). Ainda no contexto de forças atrativas, introduzimos um modelo de mistura binária de objetos intrinsecamente uniaxiais e objetos intrinsecamente biaxiais, numa formulação annealed. Essa mistura apresenta diagramas de fases bastante ricos, com topologias diversas, onde identificamos estruturas uniaxiais e biaxiais, fases reentrantes e uma grande variedade de pontos multicríticos (tricríticos, pontos críticos terminais, etc.). No caso de interações estéricas, estudamos uma teoria do funcional densidade para sistemas anisotrópicos densos, construída a partir de uma aproximação de van der Waals. Para esferoides prolatos, o modelo prevê um espaço de orientações com regiões não-acessíveis para as partículas. Além disso, o sistema apresenta uma região de coexistência entre as fases nemática e isotrópica. / We study, in a mean-field approximation, microscopic models which can lead to nematic liquid-crystalline phases. Considering attractive forces, we investigate models with quadrupolar interactions for intrinsically biaxial objects. These models present uniaxial and biaxial nematic mesophases, triple and multicritical points (tricritical point, Landau point, etc.). We also introduce a model for a binary mixture of intrinsically uniaxiail and biaxial objects, in an annealed treatment. The mixture exhibits phase diagrams with very rich topologies, where we find uniaxial and biaxial structures, reentrant phases and many different multicritical behaviors (tricritical point, critical endpoint, etc.). Moreover, assuming steric interactions, we investigate a density functional theory for hard anisotropic bodies at high densities, based on a van der Waals approximation. For hard spheroids, the model leads to an orientation space with forbidden regions for the particles. Also, the system phase separates in a nematic and an isotropic phases.

Este trabalho apresenta uma extensiva série de resultados experimentais da condutividade induzida por radiação X contínua, durante e após a irradiação, em amostras de PET e Kapton. As medidas foram realizadas variando-se uma série de parâmetros, tais como: o campo elétrico aplicado, a taxa de exposição, a espessura da amostra, o tipo de eletrodo, a energia da radiação e as condições ambientes. Foram feitas ainda medidas da corrente termo-estimulada em amostras irradiadas e não irradiadas, que permitiram verificar a presença de armadilhas nos materiais. Medidas da corrente fotônica com diferentes eletrodos e espessuras das amostras, constataram a influência do eletrodo no valor dessa corrente. Finalmente um modelo teórico foi desenvolvido, baseado na teoria de balanço dos portadores generalizada, com a inclusão do efeito do campo elétrico na taxa de geração de portadores (efeito Onsager). O ajuste teórico-experimental permitiu a determinação numérica dos principais parâmetros de condução, tais como, mobilidade dos portadores, coeficiente de recombinação e densidade de armadilhas, para os dois materiais estudados. / In this work we present extensive results of measurements of the prompt and delayed radiation-induced conductivity of samples of PET and Kapton. Experimental parameters, such as the effective energy of the radiation, the exposure rate, the total dose, the value of the applied electric field, the nature of the electrodes, and the ambienta1 conditions were changed within wide limits. We also report measurement of thermally stimulated currents for non-irradiated and for irradiated samples which allowed us to investigate the trap-structure of the materials. Measurements of photo-Compton currents with different electrode materials and sample thicknesses gave information about the relation between the nature of the electrodes and the amplitudes of the currents. Based on the generalized rate theory of radiation-induced conduction we developed a theoretical model which includes the effect of the applied electric field on the carrier generation yield (geminate recombination, Onsager effect). Comparison of experimental and theoretical curves allowed us to determine the values of the main conduction parameters, such as carrier mobility, recombination coefficient, trap densities, for the materials under investigation.

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the 3-D incompressible Euler equations belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve kinetic energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which do not conserve kinetic energy. The first part, relating to conservation of kinetic energy, has since been confirmed (cf. Eyink 1994, Constantin-E-Titi 1994). The second part, relating to the existence of non-conservative solutions, remains an open conjecture and is the subject of this dissertation. In groundbreaking work of De Lellis and Székelyhidi Jr. (2012), the authors constructed the first examples of non-conservative Hölder continuous weak solutions to the Euler equations. The construction was subsequently improved by Isett (2012/2013), introducing many novel ideas in order to construct 1/5− Hölder continuous weak solutions with compact support in time. Adhering more closely to the original scheme of De Lellis and Székelyhidi Jr., we present a comparatively simpler construction of 1/5− Hölder continuous non-conservative weak solutions which may in addition be made to obey a prescribed kinetic energy profile. Furthermore, we extend this scheme in order to construct weak non-conservative solutions to the Euler equations whose Hölder 1/3− norm is Lebesgue integrable in time. The dissertation will be primarily based on three papers, two of which being in collaboration with De Lellis and Székelyhidi Jr.