In dieser Dissertation werden das Anregungsspektrum und das Phasendiagramm wechselwirkender Fermi- und Bosegase untersucht. Zu diesem Zweck wird eine neuartige renormierte Kadanoff-Martin-Näherung vorgestellt, die Selbstwechselwirkung von Teilchen vermeidet und somit eine einheitliche Beschreibung sowohl der normalen als auch der kondensierten Phase ermöglicht. Für Fermionen findet man den BCS-Zustand, benannt nach Bardeen, Cooper und Schrieffer, welcher entscheidend ist für das Phänomen der Supraleitung. Charakteristisch für diesen Zustand ist eine Energielücke im Anregungsspektrum an der Fermi-Energie. Weiterhin tritt für Bosonen eine Bose-Einstein-Kondensation (BEC) auf, bei der das Anregungsspektrum für kleine Impulse linear ist. Letzteres führt zum Phänomen der Suprafluidität. Über die bereits bekannten Phänomene hinaus findet man eine dem BCS-Zustand ähnliche Kondensation von Zweiteilchenbindungszuständen, sowohl für Fermionen als auch für Bosonen. Für Fermionen tritt ein Übergang zwischen der Kondensation von Bindungszuständen und dem BCS-Zustand auf, der sogenannte BEC-BCS-Übergang. Die Untersuchung der Zustandsgleichung zeigt, dass im Gegensatz zu Fermi-Gasen und Bose-Gasen mit abstoßender Wechselwirkung Bose-Gase mit anziehender Wechselwirkung zu einer Flüssigkeit kondensieren oder sich verfestigen, bevor es zur Kondensation von Bindungszuständen oder zur Bose-Einstein-Kondensation kommt. Daher können diese Phänomene voraussichtlich nicht in der Gasphase beobachtet werden. Zusammenfassend lässt sich sagen, dass das vorgestellte Näherungsverfahren sehr gut geeignet ist, die erwähnten Phänomene im Zusammenhang mit der Bose-Einstein-Kondensation zu beschreiben.

Le phénomène de condensation d’un gaz de bosons lorsqu’il est refroidi à zéro degrés Kelvin futdécrit par Einstein en 1925 en s’appuyant sur des travaux de Bose. Depuis lors, de nombreux physiciens,mathématiciens et numériciens se sont intéressés au condensat de Bose-Einstein et à son caractère superfluide. Nous proposons dans cette étude des méthodes numériques ainsi qu’un code informatique pour la simulation d’un condensat de Bose-Einstein en rotation. Le principal modèle mathématique décrivant ce phénomène physique est une équation de Schrödinger présentant une non-linéarité cubique,découverte en 1961 : l’équation de Gross-Pitaevskii (GP). En nous appuyant sur le logiciel FreeFem++,nous nous servons d’une discrétisation spatiale en éléments-finis pour résoudre numériquement cette équation. Une méthode d’adaptation du maillage à la solution et l’utilisation d’éléments-finis d’ordre deux nous permet de résoudre finement le problème et d’explorer des configurations complexes en deux ou trois dimensions d’espace. Pour sa version stationnaire, nous avons développé une méthode de gradient de Sobolev ou une méthode de point intérieur implémentée dans la librairie Ipopt. Pour sa version instationnaire, nous utilisons une méthode de Time-Splitting combinée à un schéma de Crank-Nicolson ou une méthode de relaxation. Afin d’étudier la stabilité dynamique et thermodynamique d’un état stationnaire, le modèle de Bogoliubov-de Gennes propose une linéarisation de l’équation de Gross-Pitaevskii autour de cet état. Nous avons élaboré une méthode permettant de résoudre ce système aux valeurs et vecteurs propres, basée sur un algorithme de Newton ainsi que sur la méthode d’Arnoldi implémentée dans la librairie Arpack. / The phenomenon of condensation of a boson gas when cooled to zero degrees Kelvin was described by Einstein in 1925 based on work by Bose. Since then, many physicists, mathematicians and digitizers have been interested in the Bose-Einstein condensate and its superfluidity. We propose in this study numerical methods as well as a computer code for the simulation of a rotating Bose-Einstein condensate.The main mathematical model describing this phenomenon is a Schrödinger equation with a cubic nonlinearity, discovered in 1961: the Gross-Pitaevskii (GP) equation. By using the software FreeFem++ and a finite elements spatial discretization we solve this equation numerically. The mesh adaptation to the solution and the use of finite elements of order two allow us to solve the problem finely and to explore complex configurations in two or three dimensions of space. For its stationary version, we have developed a Sobolev gradient method or an internal point method implemented in the Ipopt library. .For its unsteady version, we use a Time-Splitting method combined with a Crank-Nicolson scheme ora relaxation method. In order to study the dynamic and thermodynamic stability of a stationary state,the Bogoliubov-de Gennes model proposes a linearization of the Gross-Pitaevskii equation around this state. We have developed a method to solve this eigenvalues and eigenvector system, based on a Newton algorithm as well as the Arnoldi method implemented in the Arpack library.

Les systèmes fortement corrélés, pouvant adopter des phases surprenantes de la matière, émergent dans le domaine des atomes ultra-froids ou dans celui de l’électrodynamique quantique en cavité (CQED). Ceux-ci sont au centre d’intenses travaux expérimentaux et théoriques. Dans cette thèse, nous présentons une étude de deux modèles de bosons avec deux ou zéro états internes. Ceux-ci peuvent se déplacer sur un réseau, et sont localement couplés avec des spins 1/2. Notre intérêt réside dans la détermination du diagramme de phase de l’état fondamental de ces systèmes ainsi que de l’étude des propriétés de phase et des transitions entre ces dernières. Nous avons utilisé deux outils : une approximation de champ moyen et des simulations de Monte-Carlo quantique, qui fournit des résultats numériquement exacts. Le premier modèle, appelé modèle de Kondo bosonique sur réseau, s’inscrit dans le contexte des atomes ultra-froids sur réseau. Nous trouvons que sa physique est proche de celle du modèle de Bose-Hubbard, présentant des phases de Mott et superfluide. Le couplage local renforce le caractère isolant et on observe l’émergence de phases magnétiques au travers de couplage direct ou indirect entre bosons et/ou spins. Les effets thermiques, inhérents à tout dispositif expériemental, sont aussi étudiés. Le second modèle s’inscrit dans le domaine de la CQED sur réseau, décrit un régime de couplage ultra-fort entre des photons et des atomes, et est appelé modèle de Rabi sur réseau. Le diagramme de phase présente juste deux phases : une phase cohérente dans laquelle les spins locaux s’ordonnent ferromagnétiquement ainsi qu’une phase incohérente compressible paramagnétique / Strongly correlated systems, where new surprising phases of matter may appear both in the context of ultra-cold atoms and cavity quantum electrodynamics, are the focus of intense experimental and theoritical activity. In this thesis we present a study of two models of bosons with two or zero internal states, that is to say spin-1/2 or spin-0 bosons. These particles can move around a lattice, and they are locally coupled to immobile spins 1/2. Our interest was to determine the ground state phase diagram, study phase properties and quantum phase transitions. We used two methods: an approximate one using a mean field approach and the other using quantum Monte-Carlo simulations, which provides numerically exact results. The first model, namely the bosonic Kondo lattice model, is in the context of ultra-cold atoms in optical lattices. We found that its physics is close to that of the Bose-Hubbard model, exhibiting Mott and superfluid phases. The local coupling strengthens the insulating behaviour of the system and magnetism emerges through indirect or direct coupling between bosons. Thermal effects, inherent in experiments, are also studied. The second model, which is in the context of light-matter interaction, describes a situation of an ultra-strong coupling between spin-0 bosons (photons) and local spins 1/2 (two levels atoms) and is known as the Rabi lattice model. The phase diagram generally consists of only two phases: a coherent phase and a compressible incoherent one. The locals

In dieser Dissertation werden das Anregungsspektrum und das Phasendiagramm wechselwirkender Fermi- und Bosegase untersucht. Zu diesem Zweck wird eine neuartige renormierte Kadanoff-Martin-Näherung vorgestellt, die Selbstwechselwirkung von Teilchen vermeidet und somit eine einheitliche Beschreibung sowohl der normalen als auch der kondensierten Phase ermöglicht. Für Fermionen findet man den BCS-Zustand, benannt nach Bardeen, Cooper und Schrieffer, welcher entscheidend ist für das Phänomen der Supraleitung. Charakteristisch für diesen Zustand ist eine Energielücke im Anregungsspektrum an der Fermi-Energie. Weiterhin tritt für Bosonen eine Bose-Einstein-Kondensation (BEC) auf, bei der das Anregungsspektrum für kleine Impulse linear ist. Letzteres führt zum Phänomen der Suprafluidität. Über die bereits bekannten Phänomene hinaus findet man eine dem BCS-Zustand ähnliche Kondensation von Zweiteilchenbindungszuständen, sowohl für Fermionen als auch für Bosonen. Für Fermionen tritt ein Übergang zwischen der Kondensation von Bindungszuständen und dem BCS-Zustand auf, der sogenannte BEC-BCS-Übergang. Die Untersuchung der Zustandsgleichung zeigt, dass im Gegensatz zu Fermi-Gasen und Bose-Gasen mit abstoßender Wechselwirkung Bose-Gase mit anziehender Wechselwirkung zu einer Flüssigkeit kondensieren oder sich verfestigen, bevor es zur Kondensation von Bindungszuständen oder zur Bose-Einstein-Kondensation kommt. Daher können diese Phänomene voraussichtlich nicht in der Gasphase beobachtet werden. Zusammenfassend lässt sich sagen, dass das vorgestellte Näherungsverfahren sehr gut geeignet ist, die erwähnten Phänomene im Zusammenhang mit der Bose-Einstein-Kondensation zu beschreiben.

Albert Einstein published the English translation of Relativity: The Special and General Theory in the midst of two big events in 1920: the confirmation of the two theories of relativity and spacetime in 1919 and the Nobel prize in physics in 1921. The new global celebrity wanted to make the theories intelligible and readable for an international English-speaking audience, an audience that also included antagonistic scientists and even anti-Semites. The aim of this thesis is to do a rhetorical analysis of Einstein’s character, his ethos, in Relativity, with a specific focus on creation of credibility in regard to his historical context: scientific ideals, values and norms as well as the political and cultural tendencies in Europe during the early 20th century. This was done firstly by identifying the implied auditor. Secondly, based on the material, I have identified three stereotypes or characters – the professional idealist, the mentor and the internationalist –  which emphases different features and capacities that are crucial for the credibility of the text. Thirdly, by using these stereotypes and in regard to the specific historical context, I investigated how Einstein developed his primary ethos into a secondary ethos in the text. The rhetorical analysis of Einstein’s Relativity shows that his ethos stands in relation to the social and cultural perception of the virtuous epistemic scientist; to fight prejudices regarding being a Jewish-German theoretical physicist; and, noteworthy, a way to produce a well-needed international space – a crucial alternative to continue the positivistic knowledge production counter to the nationalistic project.

The theory of Brownian motion is a cornerstone of modern physics. In this thesis, we introduce a nonequilibrium extension to this theory, namely an effective Markovian theory of the Brownian motion of a heated nanoparticle. This phenomenon belongs to the class of nonequilibrium steady states (NESS) and is characterized by spatially inhomogeneous temperature and viscosity fields extending in the solvent surrounding the nanoparticle. The first chapter provides a pedagogic introduction to the subject and a concise summary of our main results and summarizes their implications for future developments and innovative applications. The derivation of our main results is based on the theory of fluctuating hydrodynamics, which we introduce and extend to NESS conditions, in the second chapter. We derive the effective temperature and the effective friction coefficient for the generalized Langevin equation describing the Brownian motion of a heated nanoparticle. As major results, we find that these parameters obey a generalized Stokes–Einstein relation, and that, to first order in the temperature increment of the particle, the effective temperature is given in terms of a set of universal numbers. In chapters three and four, these basic results are made explicit for various realizations of hot Brownian motion. We show in detail, that different degrees of freedom are governed by distinct effective parameters, and we calculate these for the rotational and translational motion of heated nanobeads and nanorods. Whenever possible, analytic results are provided, and numerically accurate approximation methods are devised otherwise. To test and validate all our theoretical predictions, we present large-scale molecular dynamics simulations of a Lennard-Jones system, in chapter five. These implement a state-of-the-art GPU-powered parallel algorithm, contributed by D. Chakraborty. Further support for our theory comes from recent experimental observations of gold nanobeads and nanorods made in the the groups of F. Cichos and M. Orrit. We introduce the theoretical concept of PhoCS, an innovative technique which puts the selective heating of nanoscopic tracer particles to good use. We conclude in chapter six with some preliminary results about the self-phoretic motion of so-called Janus particles. These two-faced hybrids with a hotter and a cooler side perform a persistent random walk with the persistence only limited by their hot rotational Brownian motion. Such particles could act as versatile laser-controlled nanotransporters or nanomachines, to mention just the most obvious future nanotechnological applications of hot Brownian motion.:1 Introduction and Overview 2 Theory of Hot Brownian Motion 3 Various Realizations of Hot Brownian Motion 4 Toy Model and Numerical Methods 5 From Experiments and Simulations to Applications 6 Conclusion and Outlook

In the Thesis, two important theoretical problems arising in the theory of one-dimensional defocusing nonlinear Schrödinger (NLS) flows are investigated analytically and numerically: (i) the resonant generation of dispersive shock waves (DSWs) in one-dimensional NLS flow past a broad repulsive penetrable barrier; and (ii) the interaction of counter-propagating DSW and a simple rarefaction wave (RW), which is referred to as the DSW refraction problem. The first problem is motivated by the recent experimental observations of dark soliton radiation in a cigar-shaped BEC by sweeping through it a localised repulsive potential; the second problem represents a dispersive-hydrodynamic counterpart of the classical gas-dynamics problem of the shock wave refraction on a RW, and, apart from its theoretical significance could also find applications in superfluid dynamics. Both problems also naturally arise in nonlinear optics, where the NLS equation is a standard mathematical model and the `superfluid dynamics of light' can be used for an all-optical modelling of BEC flows. The main results of the Thesis are as follows: (i) In the problem of the transcritical flow of a BEC through a wide repulsive penetrable barrier an asymptotic analytical description of the arising wave pattern is developed using the combination of the localised ``hydraulic'' solution of the 1D Gross-Pitaevskii (GP) equation with repulsion (the defocusing NLS equation with an added external potential) and the appropriate exact solutions of the Whitham-NLS modulation equations describing the resolution of the upstream and downstream discontinuities through DSWs. We show that the downstream DSW effectively represents the train of dark solitons, which can be associated with the excitations observed experimentally by Engels and Atherton (2008). (ii) The refraction of a DSW due to its head-on collision with the centred RW is considered in the frameworks of two one-dimensional defocusing NLS models: the standard cubic NLS equation and the NLS equation with saturable nonlinearity, the latter being a standard model for the light propagation through photorefractive optical crystals. For the cubic nonlinearity case we present a full asymptotic description of the DSW refraction by constructing appropriate exact solutions of the Whitham modulation equations in Riemann invariants. For the NLS equation with saturable nonlinearity, whose modulation system does not possess Riemann invariants, we take advantage of the recently developed method for the DSW description in non-integrable dispersive systems to obtain key parameters of the DSW refraction. In both problems, we undertake a detailed analysis of the flow structure for different parametric regimes and calculate physical quantities characterising the output flows in terms of relevant input parameters. Our modulation theory analytical results are supported by direct numerical simulations of the corresponding full dispersive initial value problems (IVP).

The sequence-dependent curvature and flexibility of DNA is critical for many biochemically important processes. However, few experimental methods are available for directly probing these properties at the base-pair level. One promising way to predict these properties as a function of sequence is to model DNA with a set of base-pair parameters that describe the local stacking of the different possible base-pair step combinations. In this dissertation research, we develop and study a computational model for predicting the diffusion coefficients of short, relatively rigid DNA fragments from the sequence and the base-pair parameters. We focus on diffusion coefficients because various experimental methods have been developed to measure them. Moreover, these coefficients can also be computed numerically from the Stokes equations based on the three-dimensional shape of the macromolecule. By comparing the predicted diffusion coefficients with experimental measurements, we can potentially obtain refined estimates of various base-pair parameters for DNA. Our proposed model consists of three sub-models. First, we consider the geometric model of DNA, which is sequence-dependent and controlled by a set of base-pair parameters. We introduce a set of new base-pair parameters, which are convenient for computation and lead to a precise geometric interpretation. Initial estimates for these parameters are adapted from crystallographic data. With these parameters, we can translate a DNA sequence into a curved tube of uniform radius with hemispherical end caps, which approximates the effective hydrated surface of the molecule. Second, we consider the solvent model, which captures the hydrodynamic properties of DNA based on its geometric shape. We show that the Stokes equations are the leading-order, time-averaged equations in the particle body frame assuming that the Reynolds number is small. We propose an efficient boundary element method with a priori error estimates for the solution of the exterior Stokes equations. Lastly, we consider the diffusion model, which relates our computed results from the solvent model to relevant measurements from various experimental methods. We study the diffusive dynamics of rigid particles of arbitrary shape which often involves arbitrary cross- and self-coupling between translational and rotational degrees of freedom. We use scaling and perturbation analysis to characterize the dynamics at time scales relevant to different classic experimental methods and identify the corresponding diffusion coefficients. In the end, we give rigorous proofs for the convergence of our numerical scheme and show numerical evidence to support the validity of our proposed models by making comparisons with experimental data. / text

Cette thèse détaille la mesure des corrélations d'intensité quantiques dans des gaz d'hélium métastable. La mesure s'est opérée sur des gaz thermiques bosoniques 4He et fermioniques 3He, ainsi que sur des condensats de Bose-Einstein.<br /><br />En 1956, Robert Hanbury Brown et Richard Twiss ont mesuré la corrélation entre des photons provenant d'une même source thermique. Ils avaient ainsi mis en évidence que les photons emis par une telle source arrivaient préférentiellement groupés sur le détecteur. Ce groupement charactérise les bosons provenant d'une source non-cohérente. Les fermions manifestent un anti-groupement dans les mêmes conditions.<br /><br />En utilisant des atomes d'hélium métastables, dont l'utilisation de galettes de micro-canuax facilite la détection individuelle, nous avons pu mettre en évidence un regroupement similaire des bosons 4He provenant de sources thermiques de l'ordre du microkelvin. La cohérence des condensats de Bose-Einstein n'a pas permis de dégager une corrélation particulière, comme attendue. Une mesure sur des gaz thermiques des fermions 3He a permis de mettre en évidence leur anti-groupement. Un soin particulier a été pris pour décrire le détecteur à base de galettes de microcanaux et de lignes à retard, une des clés de la réussite de la mesure.

The thesis is divided into two loosely connected parts: the first one is concerned with three dimensional Topologically massive gravity (TMG) and the other is devoted to generating solutions of black objects within five minimal dimensional supergravity theory (mSUGRA5).