La théorie de la dynamique des processus chimiques élementaires cherche à décrire quantitativement les collisions réactives à l'échelle atomique. Les mouvements des noyaux étant extrêmement difficiles à traiter dans le formalisme quantique, les tomes sont souvent considérés comme des objets classiques. Cepandant, les effets purement quantiques jouent un rôle majeur dans certaines situations, alors que la description classique les néglige. Cette thèse apporte de nouvelles perspectives sur l'inclusion, dans le formalisme clasique, de forts effets quantiques, à savoir la quantification des mouvements internes des réactifs et produits. / The goal of chemical reaction dynamics theory is the quantitative description of reactive molecular collistions at the atomic scale. Since nuclear motions are difficult to study quantum mechanically, nuclei are often considered as classical object. However, quantum effects may play a major role in some situation, and the standard classical description does not take them into account. This thesis brings new perspectives on the inclusion into the classical treatment of one of the strongest qunatum effects, the quantization of reagents and products.

This thesis contains several theoretical studies on optomechanical systems, i.e. physical devices where mechanical degrees of freedom are coupled with optical cavity modes. This optomechanical interaction, mediated by radiation pressure, can be exploited for cooling and controlling mechanical resonators in a quantum regime. The goal of this thesis is to propose several new ideas for preparing meso- scopic mechanical systems (of the order of 10^15 atoms) into highly non-classical states. In particular we have shown new methods for preparing optomechani-cal pure states, squeezed states and entangled states. At the same time, proce-dures for experimentally detecting these quantum effects have been proposed. In particular, a quantitative measure of non classicality has been defined in terms of the negativity of phase space quasi-distributions. An operational al- gorithm for experimentally estimating the non-classicality of quantum states has been proposed and successfully applied in a quantum optics experiment. The research has been performed with relatively advanced mathematical tools related to differential equations with periodic coefficients, classical and quantum Bochner’s theorems and semidefinite programming. Nevertheless the physics of the problems and the experimental feasibility of the results have been the main priorities. / Die vorliegende Arbeit besteht aus verschiedenen theoretischen Untersuchungen von optomechanischen Systemen, das heißt physikalische Bauteile bei denen mechanische Freiheitsgrade mit Lichtmoden in optischen Kavitäten gekoppelt sind. Diese optimechanischen Wechselwirkungen, die über den Strahlungsdruck vermittelt werden, lassen sich zur Kühlung und Kontrolle von mechanischen Resonatoren im Quantenregime verwenden. Das Ziel dieser Arbeit ist es, verschiedene neue Ideen für Methoden vorzuschlagen, mit denen sich mesoskopische mechanische Systeme (bestehend aus etwa 10^15 Atomen) in sehr nicht-klassischen Zuständen präparieren lassen. Außerdem werden Techniken beschrieben, mit denen sich diese Quateneffekte experimentell beobachten lassen. Insbesondere wird ein quantitatives Maß für Nichtklassizität auf der Basis von Quasiwahrscheinlichkeitsverteilungen im Phasenraum definiert und ein operationeller Algorithmus zu dessen experimenteller Beschrieben, der bereits erfolgreich in einem quantenoptischen Experiment eingesetzt wurde.

Dans la première partie de cette thèse nous étudions la propagation des mesures de Wigner associées aux solutions de l'équation de Schrödinger à potentiels présentant des singularités coniques, et nous montrons qu'elles sont transportées par deux différents flots Hamiltoniens, l'un sur le fibré cotangent à la variété des singularités et l'autre ailleurs dans l'espace des phases, à moins d'un phénomène d'échange entre ces deux régimes qui peut se produire quand des trajectoires du flot extérieur atteignent le fibré cotangent. Nous décrivons en détail et le flot et la concentration de masse autour et sur la variété singulière, et illustrons avec des exemples quelques questions issues de la faute d'unicité des trajectoires classiques sur les singularités en dépit de l'unicité des solutions quantiques, ce qui refute tout principe de sélection classique, mais qui n'empêche dans certains cas de résoudre complètement le problème.Dans la deuxième partie nous présentons un travail mené en collaboration avec Dr. Clotilde Fermanian et Dr. Fabricio Macià où nous analysons une équation de type Schrödinger pertinente à l'étude semiclassique de la dynamique d'un électron dans un cristal avec impuretés et montrons que, dans la limite où la période caractérisique du réseau cristallin est sufisamment petite par rapport à la variation du potentiel extérieur représentant les impuretés, cette équation peut être approximée par une équation de masse effective, ou, plus généralement, que sa solution se décompose en modes de Bloch et que chacun d'eux satisfait une équation de masse effective spécifique à son énergie de Bloch / In the first part of this thesis we study the propagation of Wigner measures linked to solutions of the Schrödinger equation with potentials presenting conical singularities and show that they are transported by two different Hamiltonian flows, one over the bundle cotangent to the singular set and the other elsewhere in the phase space, up to a transference phenomenon between these two regimes that may arise whenever trajectories in the outsider flow lead in or out the bundle. We describe in detail either the flow and the mass concentration around and on the singular set and illustrate with examples some issues raised by the lack of uniqueness for the classical trajectories on the singularities despite the uniqueness of quantum solutions, dismissing any classical selection principle, but in some cases being able to fully solve the problem.In the second part we present a work in collaboration with Dr. Clotilde Fermanian and Dr. Fabricio Macià where we analyse a Schrödinger-like equation pertinent to the semiclassical study of the dynamics of an electron in a crystal with impurities, showing that in the limit where the characteristic lenght of the crystal's lattice can be considered sufficiently small with respect to the variation of the exterior potential modelling the impurities, then this equation is approximated by an effective mass equation, or, more generally, that its solution decomposes in terms of Bloch modes, each of them satisfying an effective mass equation specificly assigned to their Bloch energies

A mecânica quântica apresenta assimetria por reversão temporal. Podemos entender que a origem de tal assimetria é a forma como a medida é feita na mecânica quântica descrita usualmente. Em 1964, Aharonov, Bergmann e Lebowitz propõem a pós-seleção, instrumento capaz de torná-la simétrica no tempo, bem como argumentam a validade física de tal. Aqui é discutida uma outra saída, utilizando Teorias Probabilístico-Operacionais. Encontra-se uma forma mais geral do Teorema de Wigner. Além disso, se aplica esse novo formalismo, que vale em espaços de Hilbert de dimensão finita, no experimento de medida fraca descrito por Aharonov, Albert e Vaidman em 1988, em que preparamos um estado térmico dentro de uma caixa, estudamos seu comportamento e realizamos a reversão temporal do mesmo, confirmando que a probabilidade se mantém invariante sob tal transformação de simetria. / Quantum mechanics presents a time-reversal asymmetry. We see the origin of such asymmetry is the way how we make measurements in the usually described quantum mechanics. In 1964, Aharonov, Bergmann and Lebowitz propose the post-selection, a tool able to make it time symmetric, as well as argues such physical validity of this tool. Here, we discuss another exit, using operational-probabilistic theories. We find a more general form of Wigner´s Theorem. Moreover, we apply this new formalism, true in finite-dimensional Hilbert spaces, in Aharonov, Albert and Vaidman 1988 weak measure described experiment, in which we prepare a thermal state inside a box, study its behaviour and perform the time reversal of it, confirming that the probability is invariant under such symmetry transformation.

A K-teoria algébrica é um ramo da álgebra que associa para cada anel com unidade R, uma sequência de grupos abelianos chamados os n-ésimos K-grupos de R. Em 1970, Daniel Quillen dá uma definição geral dos K-grupos de um anel qualquer R a partir da +-construção do espaço classificante BGL(R). Por outro lado, considerando R um anel comutativo, obtém-se também a definição dos K-grupos de Milnor KMn (R). Usando o produto dos K-grupos de Quillen e Milnor e suas estruturas anti-comutativas, definimos o seguinte homomorfismo tn : KMn (R) &rarr; Kn(R): Mostraremos nesta dissertação que se R é um anel local com ideal maximal m tal que R / m é um corpo infinito, então esse homomorfismo é um isomorfismo para 0 &le; n &le; 2. Em geral tn nem sempre é injetor ou sobrejetor. Por exemplo quando n = 3, sabe-se que t3 não é sobrejetor e definimos a parte indecomponível de K3(R) como sendo o grupo Kind3 (R) := coker (KM3 (R) &rarr; t3 K3(R)). Usando alguns resultados de homologia dos grupos lineares, nesta dissertação mostraremos a existência da sequência exata de Bloch-Wigner para corpos infinitos. Esta sequência dá uma descrição explícita da parte indecomponível do terceiro K-grupo de um corpo infinito. TEOREMA (Sequência exata de Bloch-Wigner). Seja F um corpo infinito e seja p(F) o grupo de pre-Bloch de F, isto é, o grupo quociente do grupo abeliano livre gerado pelos símbolos [a], a &isin; F×, pelo subgrupo gerado por elementos da forma [a] - [b] + [b/a] - [1-a-1 /1-b-1] + [1-a /1-b] com a, b &isin; F× - {1}, a /= b. Então temos a sequência exata TorZ1 (&mu; (F), &mu; (F)) ~ &rarr; Kind3 (F) &rarr; p(F) &rarr; (F× &#8855; ZFx)&sigma; F×)&sigma; &rarr; K2(F) &rarr; 0 onde (F× &#8855; ZF×)&sigma; := (F×; &#8855; ZF×)/<a &#8855; b + b &#8855; a | a, b &isin; F×> e TorZ1 (&mu; (F); &mu; (F)) ~ é a única extensão não trivial de Z=2Z por TorZ1 (&mu; (F); &mu; (F)) se char(F) &ne; 2 e &mu; 2 &infin; (F) é finito e é TorZ1 (&mu; (F); &mu; (F)) caso contrário. O homomorfismo p(F) &rarr; (F× &#8855; ZF×) &sigma; é definido por [a] &rarr; a &#8855; (1-a). O estudo da sequência exata de Bloch-Wigner é justificada pela relação entre o segundo e terceiro K-grupo de um corpo F. / The algebraic K-theory is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +-construction of the classifying space BGL(R). On the other hand, if we consider a commutative ring R, we can define the Milnors K-groups, KMn (R), of R. Using the product of the Quillen and Milnors K-groups and their anti-commutative structure, we define a natural homomorphism tn : KMn (R) &rarr; Kn(R): In this dissertation, we show that if R is a local ring with maximal ideal m such that R=m is infinite, then this map is an isomorphism for 0<= n<= 2. But in general tn is not injective nor is surjective. For example when n = 3, we know that t3 is not surjective and define the indecomposable part of K3(R) as the group Kind3 (R) := coker (KM3 (R) &rarr; t3 K3(R)). Using some results about the homology of linear groups, in this dissertation we will prove the Bloch-Wigner exact sequence over infinite fields. This exact sequence gives us a precise description of the indecomposable part of the third K-group of an infinite field. THEOREM (Bloch-Wigner exact sequence). Let F be an infinite field and let p(F) be the pre-Bloch group of F, that is, the quotient group of the free abelian group generated by symbols [a], a &isin; F× - [1}, by the subgroup generated by the elements of the form [a][b]+ b/a][ 1-a-1/1-b-1]+ [1-a/1-b] with a; b &isin; F×, a =/ b. Then we have the exact sequence TorZ1 (&mu; (F), &mu; (F)) ~ &rarr; Kind3 (F) &rarr; p(F) &rarr; (F× &#8855; ZF×)$sigma; &rarr; K2(F) &rarr; 0 where (F× &#8855; ZF×)&sigma; := (F× &#8855; ZF×) / a &38855; b +b &#8855; a | a; b &isin; F× and TorZ1(&mu;(F);&mu;(F)) is the unique non trivial extension of Z=2Z by TorZ1 (&mu; (F); &mu; (F)) if char(F) =/ 2 and &mu;2 &infin; is finite and is TorZ1 (&mu; (F);&mu; (F)) otherwise. The homomorphism p(F) &rarr; (F×ZF×)%sigma; is defined by [a] &rarr; a &#8855; (1-a). As it is shown, the study of the Bloch-Wigner exact sequence is also justified by the relation between the second and third K-group of a field F.

Brushless Direct Current (BLDC) motors are one of the motor types rapidly gaining popularity. BLDC motors are being increasingly used in critical high performance industries such as appliances, automotive, aerospace, consumer, medical, industrial automation equipment and instrumentation. Fault detection and condition monitoring of BLDC machines is therefore assuming a new importance. The objective of this research is to advance the field of rotor and load fault diagnosis in BLDC machines operating in a variety of operating conditions ranging from constant speed to continuous transient operation. This objective is addressed as three parts in this research. The first part experimentally characterizes the effects of rotor faults in the stator current and voltage of the BLDC motor. This helps in better understanding the behavior of rotor defects in BLDC motors. The second part develops methods to detect faults in loads coupled to BLDC motors by monitoring the stator current. As most BLDC applications involve non-stationary operating conditions, the diagnosis of rotor faults in non-stationary conditions forms the third and most important part of this research. Several signal processing techniques are reviewed to analyze non-stationary signals. Three new algorithms are proposed that can track and detect rotor faults in non-stationary or transient current signals.

A K-teoria algébrica é um ramo da álgebra que associa para cada anel com unidade R, uma sequência de grupos abelianos chamados os n-ésimos K-grupos de R. Em 1970, Daniel Quillen dá uma definição geral dos K-grupos de um anel qualquer R a partir da +-construção do espaço classificante BGL(R). Por outro lado, considerando R um anel comutativo, obtém-se também a definição dos K-grupos de Milnor KMn (R). Usando o produto dos K-grupos de Quillen e Milnor e suas estruturas anti-comutativas, definimos o seguinte homomorfismo tn : KMn (R) &rarr; Kn(R): Mostraremos nesta dissertação que se R é um anel local com ideal maximal m tal que R / m é um corpo infinito, então esse homomorfismo é um isomorfismo para 0 &le; n &le; 2. Em geral tn nem sempre é injetor ou sobrejetor. Por exemplo quando n = 3, sabe-se que t3 não é sobrejetor e definimos a parte indecomponível de K3(R) como sendo o grupo Kind3 (R) := coker (KM3 (R) &rarr; t3 K3(R)). Usando alguns resultados de homologia dos grupos lineares, nesta dissertação mostraremos a existência da sequência exata de Bloch-Wigner para corpos infinitos. Esta sequência dá uma descrição explícita da parte indecomponível do terceiro K-grupo de um corpo infinito. TEOREMA (Sequência exata de Bloch-Wigner). Seja F um corpo infinito e seja p(F) o grupo de pre-Bloch de F, isto é, o grupo quociente do grupo abeliano livre gerado pelos símbolos [a], a &isin; F×, pelo subgrupo gerado por elementos da forma [a] - [b] + [b/a] - [1-a-1 /1-b-1] + [1-a /1-b] com a, b &isin; F× - {1}, a /= b. Então temos a sequência exata TorZ1 (&mu; (F), &mu; (F)) ~ &rarr; Kind3 (F) &rarr; p(F) &rarr; (F× &#8855; ZFx)&sigma; F×)&sigma; &rarr; K2(F) &rarr; 0 onde (F× &#8855; ZF×)&sigma; := (F×; &#8855; ZF×)/<a &#8855; b + b &#8855; a | a, b &isin; F×> e TorZ1 (&mu; (F); &mu; (F)) ~ é a única extensão não trivial de Z=2Z por TorZ1 (&mu; (F); &mu; (F)) se char(F) &ne; 2 e &mu; 2 &infin; (F) é finito e é TorZ1 (&mu; (F); &mu; (F)) caso contrário. O homomorfismo p(F) &rarr; (F× &#8855; ZF×) &sigma; é definido por [a] &rarr; a &#8855; (1-a). O estudo da sequência exata de Bloch-Wigner é justificada pela relação entre o segundo e terceiro K-grupo de um corpo F. / The algebraic K-theory is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +-construction of the classifying space BGL(R). On the other hand, if we consider a commutative ring R, we can define the Milnors K-groups, KMn (R), of R. Using the product of the Quillen and Milnors K-groups and their anti-commutative structure, we define a natural homomorphism tn : KMn (R) &rarr; Kn(R): In this dissertation, we show that if R is a local ring with maximal ideal m such that R=m is infinite, then this map is an isomorphism for 0<= n<= 2. But in general tn is not injective nor is surjective. For example when n = 3, we know that t3 is not surjective and define the indecomposable part of K3(R) as the group Kind3 (R) := coker (KM3 (R) &rarr; t3 K3(R)). Using some results about the homology of linear groups, in this dissertation we will prove the Bloch-Wigner exact sequence over infinite fields. This exact sequence gives us a precise description of the indecomposable part of the third K-group of an infinite field. THEOREM (Bloch-Wigner exact sequence). Let F be an infinite field and let p(F) be the pre-Bloch group of F, that is, the quotient group of the free abelian group generated by symbols [a], a &isin; F× - [1}, by the subgroup generated by the elements of the form [a][b]+ b/a][ 1-a-1/1-b-1]+ [1-a/1-b] with a; b &isin; F×, a =/ b. Then we have the exact sequence TorZ1 (&mu; (F), &mu; (F)) ~ &rarr; Kind3 (F) &rarr; p(F) &rarr; (F× &#8855; ZF×)$sigma; &rarr; K2(F) &rarr; 0 where (F× &#8855; ZF×)&sigma; := (F× &#8855; ZF×) / a &38855; b +b &#8855; a | a; b &isin; F× and TorZ1(&mu;(F);&mu;(F)) is the unique non trivial extension of Z=2Z by TorZ1 (&mu; (F); &mu; (F)) if char(F) =/ 2 and &mu;2 &infin; is finite and is TorZ1 (&mu; (F);&mu; (F)) otherwise. The homomorphism p(F) &rarr; (F×ZF×)%sigma; is defined by [a] &rarr; a &#8855; (1-a). As it is shown, the study of the Bloch-Wigner exact sequence is also justified by the relation between the second and third K-group of a field F.

Le calcul de quantités dépendants du temps pour des systèmes quantiques est limité le scaling exponentiel des méthodes exactes. Néanmoins, ces quantités présentes un intérêt scientifique important. Un compromis, entre précision et coût, est trouvé par les méthodes quasi-classiques. Dans ces méthodes, la densité thermique exacte est combinée à des trajectoires approximant la dynamique quantique. Durant ma thèse, j’ai développé et appliqué une méthode quasi-classique : PIM (Phase Integration Methode) qui combine des algorithmes MC et MD pour calculer des fonction de corrélation. Le Chapitre 2 décrit les méthodes quasi-classiques ainsi que les approximations qui permettent d’en tirer les fonctions de corrélations dépendants du temps.Le Chapitre 3 illustre comment PIM est adapté au calcul de la densité de Wigner qui est une quantité clé pour les méthodes quasi-classique. À travers le calcul de cette quantité, PIM est capable de capturer des corrélations entre différents degrés de liberté. Dans le Chapitre 4, on montre comment PIM est adapté au calcul de spectres infrarouge. La comparaison des résultats avec d’autres méthodes montre que PIM est une méthode précise pour les systèmes à basse dimensionnalité. Les spectres de OH et CH4 confirment que PIM ne souffrent pas de problèmes intrinsèques comme CMD ou RPMD et peut être appliqué à des systèmes à plus haute dimensionnalité. Le Chapitre 5 présente la méthodologie pour calculer des constantes de vitesse à l’aide de PIM. Les résultats sont bons jusqu’à 300 K mais pas en dessous. Le travail futur se concentrera sur le calcul de la fonction de corrélation de Kubo flux-side pour remédier à ce problème. / Simulation of time-dependent quantities for quantum systems is limited by the exponential scaling of exact methods. However, the calculation of these quantities is key in many problems. A reasonable compromise among accuracy and cost is done by the quasi-classical methods for computing time correlation functions. In these methods, the thermal density is combined with trajectories that approximate quantum dynamics. In my thesis, I develop and apply quasi-classical methods for vibrational spectroscopy. The focus is on the Phase Integration Method. PIM is based on combining MD and MC algorithms to compute appropriate quantities. Chapter 2 is devoted to a general description of the quasi-classical methods. We introduce the different approximations used to compute quantum time correlation functions. Chapter 3 illustrates how PIM is adapted to the calculation of the Wigner density, which is a key quantity in quasi-classical methods. Via this quantity, we show that PIM is able to capture quantum correlation effects among different degrees of freedom. Chapter 4 focuses on the adaptation of PIM for the infrared spectroscopy. Comparison of our results, show that PIM is accurate for low dimensional models. OH and CH4 spectrum confirms that our approach does not suffer from the pathologies such as CMD and RPMD but also that it can treat systems with a larger number of degrees of freedom reliably. Chapter 5 presents the methodology used to calculate rate constants with PIM. The results are in good agreement with the exact reference until 300 K. Future work will focus on using the Kubo flux side correlation function.

Orientador: Guillermo Gerardo Cabrera Oyarzún / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-30T18:12:28Z (GMT). No. of bitstreams: 1 Hernandez_FabioHernandez_M.pdf: 2178317 bytes, checksum: 00e18623b835112b2aa5c348e4651b65 (MD5) Previous issue date: 2016 / Resumo: A descrição da dinâmica quântica de sistemas de muitos corpos é um ingrediente chave para computação e simulações quânticas. No presente projeto, estudamos a dinâmica de cadeias de spin na presença de impurezas ou defeitos. O sistema de Ising quantico (Ising com campo transverso) com uma impureza foi solucionado de forma exata. Este sistema de spins pode ser simulado de forma analítica por partículas quânticas (transformação de Jordan-Wigner). Caracterizamos o espectro, as autofunções e a evolução temporal da magnetização para estados iniciais particulares, focando no papel desempenhado pelos estados de impureza. Finalmente observamos oscilações remanescentes na magnetização, após a relaxação do sistema, para alguns valores dos parâmetros da impureza nos quais existem dois estados ligados no espectro de energias / Abstract: The description of dynamics of quantum many-body systems is a key ingredient to perform quantum computation and/or simulations of quantum behavior. In the present proposal, we study the time evolution of quantum spin chains with impurities at one of the boundaries, in order to understand the role of defects in relaxation properties. The quantum (transverse) Ising model with an impurity has been solved in exact form, using the Jordan-Wigner transformation, where spins are mapped onto spinless fermions, thus simulating analytically a spin system with particles. We completely characterize the spectrum, with the presence of bound states depending on values of the impurity parameters. We calculate the local magnetization and observe its relaxation for particular non-homogeneous initial states. Surprisingly, remanent Rabi oscillations are observed at asymptotically long times, when the spectrum displays two bound states / Mestrado / Física / Mestre em Física / 1247646/2013 / CAPES

Cette thèse est dédiée à l'étude mathématique de l'approximation de champ-moyen des gaz de bosons. En physique quantique une telle approximation est vue comme la première approche permettant d'expliquer le comportement collectif apparaissant dans les systèmes quantiques à grand nombre de particules et illustre des phénomènes fondamentaux comme la condensation de Bose-Einstein et la superfluidité. Dans cette thèse, l'exactitude de l'approximation de champ-moyen est obtenue de manière générale comme seule conséquence de principes de symétries et de renormalisations d'échelles. Nous recouvrons l'essentiel des résultats déjà connus sur le sujet et de nouveaux sont prouvés, particulièrement pour les systèmes quantiques sur réseau, incluant le modèle de Bose-Hubbard. D'autre part, notre étude établit un lien entre les équations aux hiérarchies de Gross-Pitaevskii et de Hartree, issues des méthodes BBGKY de la physique statistique, et certaines équations de transport ou de Liouville dans des espaces de dimension infinie. Résultant de cela, les propriétés d'unicité pour de telles équations aux hiérarchies sont prouvées en toute généralité utilisant seulement les caractéristiques génériques de problèmes aux valeurs initiales liés à de telles équations. Egalement, de nouveaux résultats de caractères bien posés et un contre-exemple à l'unicité d'une hiérarchie de Gross-Pitaevskii sont prouvés. L’originalité de nos travaux réside dans l'utilisation d'équations de Liouville et de puissantes techniques de transport étendues à des espaces fonctionnels de dimension infinie et jointes aux mesures de Wigner, ainsi qu'à une approche utilisant les outils de la seconde quantification. Notre contribution peut être vue comme l'aboutissement d'idées initiées par Z. Ammari, F. Nier et Q. Liard autour de la théorie de champ-moyen. / This thesis is dedicated to the mathematical study of the mean-field approximation of Bose gases. In quantum physics such approximation is regarded as the primary approach explaining the collective behavior appearing in large quantum systems and reflecting fundamental phenomena as the Bose-Einstein condensation and superfluidity. In this thesis, the accuracy of the mean-field approximation is proved in full generality as a consequence only of scaling and symmetry principles. Essentially all the known results in the subject are recovered and new ones are proved specifically for quantum lattice systems including the Bose-Hubbard model. On the other hand, our study sets a bridge between the Gross-Pitaevskii and Hartree hierarchies related to the BBGKY method of statistical physics with certain transport or Liouville's equations in infinite dimensional spaces. As an outcome, the uniqueness property for these hierarchies is proved in full generality using only generic features of some related initial value problems. Again, several new well-posedness results as well as a counterexample to uniqueness for the Gross-Pitaevskii hierarchy equation are proved. The originality in our works lies in the use of Liouville's equations and powerful transport techniques extended to infinite dimensional functional spaces together with Wigner probability measures and a second quantization approach. Our contributions can be regarded as the culmination of the ideas initiated by Z. Ammari, F. Nier and Q. Liard in the mean-field theory.