The statistical analysis of cointegration is crucial for inferring shared stochastic trends between variables and is an important area of Econometrics for analyzing long-term equilibriums in the economy. Bayesian inference of cointegration involves the identification of cointegrating vectors that are determined up to arbitrary linear combinations, for which the Gibbs sampler is often used to simulate draws from the posterior distribution. However, economic theory may not suggest linear relations and regime-switching models can be used to account for non-linearity. Modeling cointegration and regime-switching as well as the combination of them are associated with highly parameterized models that can prove to be difficult for Markov Chain Monte Carlo techniques such as the Gibbs sampler. Hamiltonian Monte Carlo, which aims at efficiently exploring the posterior distribution, may thus facilitate these difficulties. Furthermore, posterior distributions with highly varying curvature in their geometries can be adequately monitored by Hamiltonian Monte Carlo. The aim of the thesis is to analyze how Hamiltonian Monte Carlo performs in simulating draws from the posterior distributions of models accounting for cointegration and regime-switching. The results suggest that while it is not necessarily the case that regime-switching will be identified, Hamiltonian Monte Carlo performs well in exploring the posterior distribution. However, high rates of divergences from the true Hamiltonian trajectory reduce the algorithm to a Random Walk to some extent, limiting the efficiency of the sampling.

Phonon anharmonicity is critical for describing various phenomena in crystals, including lattice thermal conductivity, thermal expansion, structural phase transitions, and many others. Including anharmonicity in the calculation of condensed matter observables developed rapidly in the past decade. First-principles computation of cubic phonon interactions have been performed in many systems, and the quartic interactions have begun to receive more attention. In this study, reliable Hamiltonians are constructed purely in terms of quadratic, cubic, and quartic irreducible derivatives, which are calculated efficiently and precisely using the lone and bundled irreducible derivative approaches (LID and BID). The resulting Hamiltonians give rise to a nontrivial many-phonon problem which requires some approximation in order to compute observables. We implemented self-consistent diagrammatic approaches to evaluate the phonon self-energy, including the Hartree-Fock approximation for phonons and quasiparticle perturbation theory, where both the 4-phonon loop and the real part of the 3-phonon bubble are employed during self-consistency. Additionally, we implemented molecular dynamics in order to yield the numerically exact solution in the classical limit. The molecular dynamics solution is robust for directly comparing to experimental results at sufficiently high temperatures, and for assessing our diagrammatic approaches in the classical limit. Anharmonic vibrational Hamiltonians were constructed for CaF₂, ThO₂, and UO₂. Diagrammatic approaches were used to evaluate the phonon self-energy, yielding the phonon lineshifts and linewidths and the thermal conductivity within the relaxation time approximation. Our systematic results allowed us to resolve the paradox of why first-principles phonon linewidths strongly disagree with results extracted from inelastic neutron scattering (INS). We demonstrated that the finite region in reciprocal space required in INS data analysis, the 𝑞-voxel, must be explicitly accounted for within the calculation in order to draw a meaningful comparison. We also demonstrated that the 𝑞-voxel is important to properly compare the spectrum measured in inelastic X-ray scattering (IXS), despite the fact that the ?-voxel is much smaller. Accounting for the 𝑞-voxel, we obtained good agreement for the scattering function linewidths up to intermediate temperatures. Additionally, good agreement was obtained for the thermal conductivity. Another topic we addressed is translation symmetry breaking caused by factors such as defects, chemical disorders, and magnetic order. These phenomena will lead to shifts and a broadening of the phonon spectrum, and formally the single-particle Green’s function encodes these effects. However, it is often desirable to obtain an approximate non-interacting spectrum that contains the effective shifts of the phonon frequencies, allowing straightforward comparison with experimentally measured scattering peak locations. Such an effective phonon dispersion can be obtained using a band unfolding technique, and in this study, we formulated unfolding in the context of irreducible derivatives. We showcased the unfolding of phonons in UZr₂, where chemical disorder is present, and compared the results with experimental IXS data. Additionally, we extended the unfolding technique to anharmonic terms and demonstrated this using 3rd and 4th order terms in the antiferromagnetic phase of UO₂.

In this thesis, newer Markov Chain Monte Carlo (MCMC) algorithms are implemented and compared in terms of their efficiency in the context of sampling from Gaussian processes under linear inequality constraints. Extending the framework of Gaussian process that uses Gibbs sampler, two MCMC algorithms, Exact Hamiltonian Monte Carlo (HMC) and Analytic Elliptical Slice Sampling (ESS), are used to sample values of truncated multivariate Gaussian distributions that are used for Gaussian process regression models with linear inequality constraints. In terms of generating samples from Gaussian processes under linear inequality constraints, the proposed methods generally produce samples that are less correlated than samples from the Gibbs sampler. Time-wise, Analytic ESS is proven to be a faster choice while Exact HMC produces the least correlated samples.

Eine Hauptanwendung der Einzelpartikel-Analyse in der Kryo-Elektronenmikroskopie ist die Charakterisierung der dreidimensionalen Struktur makromolekularer Komplexe. Dazu werden zehntausende Bilder verwendet, die verrauschte zweidimensionale Projektionen des Partikels zeigen. Im ersten Schritt werden ein niedrig aufgelöstetes Anfangsmodell rekonstruiert sowie die unbekannten Bildorientierungen geschätzt. Dies ist ein schwieriges inverses Problem mit vielen Unbekannten, einschließlich einer unbekannten Orientierung für jedes Projektionsbild. Ein gutes Anfangsmodell ist entscheidend für den Erfolg des anschließenden Verfeinerungsschrittes. Meine Dissertation stellt zwei neue Algorithmen zur Rekonstruktion eines Anfangsmodells in der Kryo-Elektronenmikroskopie vor, welche auf einer groben Darstellung der Elektronendichte basieren. Die beiden wesentlichen Beiträge meiner Arbeit sind zum einen das Modell, welches die Elektronendichte darstellt, und zum anderen die neuen Rekonstruktionsalgorithmen. Der erste Hauptbeitrag liegt in der Verwendung Gaußscher Mischverteilungen zur Darstellung von Elektrondichten im Rekonstruktionsschritt. Ich verwende kugelförmige Mischungskomponenten mit unbekannten Positionen, Ausdehnungen und Gewichtungen. Diese Darstellung hat viele Vorteile im Vergleich zu einer gitterbasierten Elektronendichte, die andere Rekonstruktionsalgorithmen üblicherweise verwenden. Zum Beispiel benötigt sie wesentlich weniger Parameter, was zu schnelleren und robusteren Algorithmen führt. Der zweite Hauptbeitrag ist die Entwicklung von Markovketten-Monte-Carlo-Verfahren im Rahmen eines Bayes'schen Ansatzes zur Schätzung der Modellparameter. Der erste Algorithmus kann aus dem Gibbs-Sampling, welches Gaußsche Mischverteilungen an Punktwolken anpasst, abgeleitet werden. Dieser Algorithmus wird hier so erweitert, dass er auch mit Bildern, Projektionen sowie unbekannten Drehungen und Verschiebungen funktioniert. Der zweite Algorithmus wählt einen anderen Zugang. Das Vorwärtsmodell nimmt nun Gaußsche Fehler an. Sampling-Algorithmen wie Hamiltonian Monte Carlo (HMC) erlauben es, die Positionen der Mischungskomponenten und die Bildorientierungen zu schätzen. Meine Dissertation zeigt umfassende numerische Experimente mit simulierten und echten Daten, die die vorgestellten Algorithmen in der Praxis testen und mit anderen Rekonstruktionsverfahren vergleichen.

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.

Ce mémoire est une poursuite de l’étude de la superintégrabilité classique et quantique dans un espace euclidien de dimension deux avec une intégrale du mouvement d’ordre trois. Il est constitué d’un article. Puisque les classifications de tous les Hamiltoniens séparables en coordonnées cartésiennes et polaires sont déjà complétées, nous apportons à ce tableau l’étude de ces systèmes séparables en coordonnées paraboliques. Premièrement, nous dérivons les équations déterminantes d’un système en coordonnées paraboliques et ensuite nous résolvons les équations obtenues afin de trouver les intégrales d’ordre trois pour un potentiel qui permet la séparation en coordonnées paraboliques. Finalement, nous démontrons que toutes les intégrales d’ordre trois pour les potentiels séparables en coordonnées paraboliques dans l’espace euclidien de dimension deux sont réductibles. Dans la conclusion de l’article nous analysons les différences entre les potentiels séparables en coordonnées cartésiennes et polaires d’un côté et en coordonnées paraboliques d’une autre côté. Mots clés: intégrabilité, superintégrabilité, mécanique classique, mécanique quantique, Hamiltonien, séparation de variable, commutation. / This thesis is a contribution to the study of classical and quantum superintegrability in a two-dimensional Euclidean space involving a third order integral of motion. It consists of an article. Because the classifications of all separable hamiltonians into Cartesian and polar coordinates are already complete, we bring to this picture the study of those systems in parabolic coordinates. First, we derive the determinating equations of a system into parabolic coordinates, after which we solve the obtained equations in order to find integrals of order three for potentials, which allow the separations of variables into the parabolic coordinates. Finally, we prove that all the third order integrals for separable potentials in parabolic coordinates in the Euclidean space of dimension two are reducible. In the conclusion of this article, we analyze the differences between the separable potentials in Cartesian and polar coordinates and the separable potentials in parabolic coordinates. Keywords: integrability, superintegrability, classical mechanics, quantum mechanics, Hamiltonian, separation of variables, commutation.

Cette thèse concerne l'ouverture de cycle des spiropyranes (SP) et leur utilisation comme interrupteurs contrôlés par impulsions laser. Nous avons élaboré un modèle théorique pour étudier cette réaction photochimique et développer des stratégies de contrôle.Les SP présentent des effets non adiabatiques impliquant un traitement quantique pour les électrons et les noyaux. L'approche multiconfigurationnelle dépendante du temps (MCTDH) est idéale car elle peut traiter des dynamiques quantiques impliquant plusieurs états électroniques. MCTDH a été appliqué avec succès aux calculs de spectres électroniques de systèmes montrant de forts effets non adiabatiques. Cependant, cette approche requiert au préalable les surfaces d'énergie potentielle (PES). Ces applications sont basées sur un modèle de couplage vibronique local de la PES à proximité du point de Franck-Condon (FC). Contrairement aux calculs de spectres de photoabsorbtion impliquant souvent de courtes échelles de temps et de faibles déformations autour du point de FC, la simulation de réactions photochimiques requiert une représentation adéquate des mouvements de large amplitude. Ce modèle doit donc être rendu plus flexible. Les modes normaux, utilisés dans le modèle standard, n'étant pas adaptés aux grandes déformations, nous avons choisi d'utiliser la méthode MCTDH en coordonnées curvilignes avec une reformulation systématique du modèle en coordonnées polysphériques afin d'obtenir une énergie cinétique nucléaire séparable.Même si le processus n'implique que l'état fondamental et le premier état excité, leurs configurations électroniques dominantes peuvent changer fortement le long de mouvements de grande amplitude. Nous avons donc développé une approche générale basée sur une étude systématique de quelques données ab initio pour générer le meilleur jeu d'états diabatiques d'un problème donné.De premières applications au benzopyrane (chromophore des SP) ont montré un bon accord avec l'expérience. / The focus of this thesis is the ring opening of spiropyrans (SP), and how these molecules can be used as photoswitches controlled with laser pulses. We have built a theoretical model to study this photochemical reaction and develop strategies for control.SP exhibit nonadiabatic effects, and therefore, their modelling requires a quantum description for both the electrons and the nuclei. The multiconfiguration time-dependent Hartree (MCTDH) approach is ideal in this case because it can treat quantum dynamics involving several electronic states. MCTDH was successfully applied to electronic spectra calculations of systems showing strong nonadiabatic effects. However, the corresponding potential energy surfaces (PES) for this approach are required first. These applications are based on a local vibronic coupling model of the PES in the neighbourhood of the Franck-Condon (FC) point.As opposed to calculating photoabsorption spectra that often involves short timescales and small deformations around the FC geometry, simulating photochemical reactions requires an adequate representation of large-amplitude motions. Thus, this model must be made more flexible.Normal modes, usually used for the standard model, are not adapted to large-amplitude deformations. We thus chose to run MCTDH in curvilinear coordinates and recast systematically the model in terms of polyspherical coordinates to produce a separable form for the nuclear kinetic energy.Even if only the ground and the first excited electronic states are involved in the process, their dominant electronic configurations may change significantly along large-amplitude motions. We have developed a general approach based on a systematic analysis of a few ab initio data to generate the best set of diabatic states for a given problem.Preliminary results applied to benzopyran (the chromophore of the SP) showed good agreement with the experiments.

L'objectif de ce thèse est l'étude des systèmes dynamiques avec interaction à longue portée. La complexité de leur dynamique met en évidence des propriétés contre-intuitives et inattendues, comme l'existence d'états stationnaires hors-équilibre (QSS). Dans le QSS on peut observer des propriétés particulières: chaleur spécifique négative, inéquivalence des ensembles statistiques et phénomènes d'auto-organisation. Les théories des interactions LR ont été appliquées pour décrire la dynamique des systèmes auto-gravitants, de tourbillons bidimensionnels, de systèmes avec interactions onde-particule et des plasmas chargés. Mon travail s'est tout d'abord consacré à l'extension de la solution de Lynden-Bell pour le modèle HMF, en généralisant l'analyse à des conditions initiales de «water-bag" à plusieurs niveaux, qui approchent des conditions initiales continues. En suite je me suis intéressé à la caractérisation formelle de la thermodynamique des QSS dans l'ensemble statistique canonique. En appliquant la théorie standard, il est possible de mesurer une chaleur spécifique "cinétique'' négative. Cette propriété inattendue amène à la violation du second principe de la thermodynamique. Un tel résultat nous pousse à reconsidérer l'applicabilité de la théorie thermodynamique actuelle aux systèmes LR. En suite j'ai étudié, pour le modèle α-HMF, la persistance des caractéristiques typiques du régime LR, dans le limite dynamique à courte portée. Les résultats suggèrent une généralisation de la définition des systèmes LR. Le dernier chapitre est consacré à la caractérisation d'un nouveau modèle LR, extension naturelle du précédent α-HMF et d'intérêt potentiel applicatif. / The scope of this thesis is the study of systems with long-range interactions (LR). The complexity of their dynamics evidences counter-intuitive and unexpected properties, as for instance the existence of out-of-equilibrium stationary states (QSS). Considering a system in the QSS, one may observe peculiar properties, as negative specific heat, statistical ensemble inequivalence and phenomena of self-organizations. The main theories of long-range interactions have been applied to describing self-gravitating systems, two-dimensional vortices, systems with wave-particle interactions and charged plasmas. My work has been initially dedicated to extending the Lynden-Bell solution for the HMF model, generalizing the analysis to multi-level water-bag initial condition that could approximate continuous distributions. Then I concentrated to the formal characterization of the thermodynamics of QSS in the canonical statistical ensemble. By applying the standard theory, it is possible to measure negative “kinetic” specific heat. This latter unexpected property leads to a violation of the second principle of thermodynamics. Such result forces us to reconsider the applicability of the accepted thermodynamic theory to LR systems. Afterwards I studied, in the context of the α-HMF model, the persistence of the typical characteristics of the LR regime in the limit of short-range dynamics. The results obtained suggests a generalization of the definition of LR systems. The last chapter is dedicated to the characterization of a novel LR model, a natural extension of α-HMF and of potential applicability.

La théorie classique des formes normales a pour but de simplifier des problèmes compliqués grâce à des changements de coordonnées réguliers pour ne conserver que les caractéristiques dynamiques du système. Plus précisément, on considère un système dynamique que l'on dit "élémentaire", comme par exemple la partie linéaire d'un champ de vecteurs au voisinage d'un point singulier, et on se donne une perturbation de ce système élémentaire. Les formes normales sont alors l'ensemble des représentants de ces perturbations à la conjugaison près d'une transformation régulière. Elles ne sont constituées que des termes qui caractérisent la dynamique du système perturbé et que l'on appelle "résonances". Dans la première partie de la thèse on cherche à comprendre la dynamique locale d'équations différentielles implicites de la forme F(x,y,y')=0, où F est un germe de fonction holomorphe au voisinage d'un point singulier. Pour cela on utilise la relation intime entre les systèmes implicites et les champs liouvilliens. La classification par transformation de contact des équations implicites provient de la classification symplectique des champs liouvilliens. On utilise alors toute la théorie des formes normales pour les champs de vecteurs, dans le cas holomorphe (Brjuno, Siegel, Stolovitch) et dans le cas réel (Sternberg), que l'on adapte pour les champs liouviliens avec des transformations symplectiques. On établit alors des résultats de classification des équations implicites en fonction des invariants dynamiques, ainsi que des conditions d'existence de solutions locales via les formes normales. / The aim of the classical theory of normal forms is to simplify complicated problems by using regular changes of coordinates, in order to keep the dynamical characteristics of the system. More precisely, we consider a dynamic system said to be "elementary", like a linear part of a vector field in the neighborhood of a singular point, and we focus on a perturbation of this elementary system. Normal forms are the set of all representatives of those perturbations under the action of the group of regular transformation. They are composed of terms which caracterise the dynamics of the perturbed system, and which are called "resonances". In the first part, we try to understand the local dynamic of implicit equations of the form $F(x,y,y')=0$, where $F$ is a germ of holomorphic function in a neighborhood of a singular point. To this end we use the relation between implicit systems and liouvillian vector fields. The classification by contact transformations of implicit equations come from the symplectic classification of liouvillian vector fields. We use all normal forms theory for vector fields, in complex case (Bjruno, Siegel, Stolovitch), and in real case (Sternberg), adapted to liouvillian fields with symplectic transformations. We establish classification results for implicit equations according to the dynamical invariants, and existence conditions of local solutions using normal forms. In the second part, we undertake the normalization of an analytic vector field in a neighborhood of the torus. Brjuno enunciates a theorem of normalization, under conditions of control of small divisors and integrability of the normal forms ; however he doesn't give any proof of that theorem.

Higher-dimensional Hamiltonian systems usually exhibit a mixed phase space in which regular and chaotic motion coexist. While regular trajectories are confined to regular tori, chaotic trajectories can be transported through a web of so called resonance channels which disrupt the regular structures. The focus of this thesis are time-discrete 4D symplectic maps which represent the lowest dimensional system for which the chaotic transport can circumvent regular tori. While the dynamics of 2D maps are well established, many fundamental questions are open for maps of dimension four and higher due to this property. In particular, the mechanism of the power-law trapping is unknown for these maps. In this thesis, the organization and hierarchy of the regular structures of 4D maps is uncovered and the slow chaotic transport close to these structures is examined. Specifically, this transport is shown to be organized by a set of overlapping resonance channels. The transport across these channels is found to be governed by partial transport barriers. For the transport along a channel a stochastic process including a drift is conjectured. Based on each of these two types of chaotic transport a possible mechanism for the power-law trapping in higher-dimensional systems is proposed.