On the general principle of the mean-field approximation for many-boson dynamics

The mean-field approximations of many-boson dynamics are known to be effective in many physical relevant situations. The mathematical justifications of such approximations rely generally on specific considerations which depend too much on the model and on the initial states of the system which are required to be well-prepared. In this article, using the method of Wigner measures, we prove in a fairly complete generality the accuracy of the mean-field approximation. Roughly speaking, we show that the dynamics of a many-boson system are well approximated, in the limit of a large number of particles, by a one particle mean-field equation if the following general principles are satisfied: • The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles). • The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part. • There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces

Mean-field theory and dynamics of lattice quantum systems

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

On well-posedness and uniqueness for general hierarchy equations of Gross-Pitaevskii and Hartree type

International audienceGross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to an initial value problem, respectively the Gross-Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this article, we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain Liouville equations and secondly solving the uniqueness and existence questions for the latter. As an outcome, we formulate a hierarchy equation starting from any initial value problem which is $U(1)$-invariant and prove a general principle which can be stated formally as follows: (i) Uniqueness for weak solutions of an initial value problem implies the uniqueness of solutions for the related hierarchy equation. (ii) Existence of solutions for the initial value problem implies existence of solutions for the related hierarchy equation. In particular, several new well-posedness results as well as a counterexample to uniqueness for the Gross-Pitaevskii hierarchy equation are proved. The novelty in our work lies in the aforementioned duality and the use of Liouville equations with powerful transport techniques extended to infinite dimensional functional spaces

On well-posedness and uniqueness for general hierarchy equations of Gross-Pitaevskii and Hartree type

International audienceGross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to an initial value problem, respectively the Gross-Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this article, we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain Liouville equations and secondly solving the uniqueness and existence questions for the latter. As an outcome, we formulate a hierarchy equation starting from any initial value problem which is $U(1)$-invariant and prove a general principle which can be stated formally as follows: (i) Uniqueness for weak solutions of an initial value problem implies the uniqueness of solutions for the related hierarchy equation. (ii) Existence of solutions for the initial value problem implies existence of solutions for the related hierarchy equation. In particular, several new well-posedness results as well as a counterexample to uniqueness for the Gross-Pitaevskii hierarchy equation are proved. The novelty in our work lies in the aforementioned duality and the use of Liouville equations with powerful transport techniques extended to infinite dimensional functional spaces