Transition vitreuse et dynamique lente.

A typical liquid can be cooled down so as to avoid crystallisation resulting in what is called a supercooled liquid. Its static properties are mere continuations of those of a liquid. However several new phenomena emerges in dynamics. The most prominent of them is the tremendous slowing down of the dynamics, referred to as a glass transition. The global relaxation time grows from a pi- cosecond scale at the crystallisation transition up to a macroscopic value for low temperatures. Understanding of the phenomena in the low temperature state of supercooled liquids, in particular the tremendous slowing down, is one of the longstanding open problems of the out of equilibrium statistical physics. Different scenarios were proposed to explain the observed low-T behaviour of supercooled liquids. In this thesis we follow the one based on the results of gen- eralised spin glass models (mean-field in nature) and the Mode-Coupling Theory (MCT). This scenario reproduces well the initial slow down of the dynamics but predicts a spurious transition to non-ergodic phase for low temperatures. Thus a problem naturally arises: how to enhance the model so as to get a correct behaviour at low temperatures and to eliminate the transition ? The main aim of the thesis is to construct such a generalisation. The general expectation is that corrections to MCT modify the low-T theory. However the main drawback of the MCT-based scenario is the uncontrolled nature of the approximation lying at the heart of MCT what makes the computation of cor- rections within the original projector operator formalism an almost unsolvable problem. A promising way to circumvent this problem is a rederivation of MCT within a field theoretical context. Revision of previous attempts reveals their inconsistency caused by the violation of the time-reversal symmetry (TRS) in perturbation series. Resolving this issue results in a correct derivation of MCT within a field theoretical context and yields perturbation series respecting TRS. This provides a direct way to test a structural stability of MCT and, thus, provides an insight on the very important problem of whether the transition cutoff can be understood within some refined approximation or has more fun- damental foundations. The result is that MCT is only a mean-field theory of the glass transition, so that perturbative corrections are irrelevant. Within this context we also explore the mapping via a Cole-Hopf transformation between the supercooled liquids and reaction-diffusion systems. The above mentioned results apply to the supercooled liquids above the transition i.e. to equilibrium dynamics. The last chapter deals with their partial generalisation to low temperatures where the system never reaches equilibrium during the observation and the global relaxation time is, effectively, infinite. Analysis turns out to be more complicated in that case since a bunch of new phenomena like ageing of physical properties comes into play.N'importe quel liquide peut etre refroidi au-dessous de sa temperature de cristallisation sans se cristalliser en devenant un liquide surfondu. Ces proprietes statiques sont les continuations analytiques de celles d'un liquide ordinaire. Par contre la dynamique montre un ralentissement dramatique qu'on apelle la transition vitreuse: le temps de relaxation croit de picoseconds au voisinage de la transition de cristallisation au temps macroscopique pour les basses temperatures. Comprehension et explication de ce comportement des liquides surfondus a basses temperatures est l'un des problemes ouverts de mecanique statistique hors d'equilibre. Les scenarios divers ont ete proposes pour expliquer ce comportement des liquides surfondus. Dans cette th`ese nous adoptons celui base sur l'analyse des modeles de verres de spins generalises et la Theorie de Couplage de Modes (MCT). Ce scenario reproduit bien le ralentissement initial de la dynamique. Cependant il predit une transition spurieuse vers une phase non-ergodique pour les basses temperatures. C a pose un probleme naturel: comment peut-t-on generaliser le modele afin de reproduire le comportement aux basses temperatures observe et eliminer la transition ? Le but principal de la these etait de construire une telle generalisation. Il est suppose de facon generale que c'est les corrections aux MCT qui modifient la theorie aux basses temperatures. Cependant le defaut principal de MCT est la nature incontrolee de l'approximation qui amene a MCT ce qui fait le calcul des corrections a MCT un probleme quasiment insoluble. Un methode promettant pour contourner cette difficulte est la rederivation de MCT dans le contexte d'une theorie de champs. La revision des essaies precedents a releve leur inconsistance a cause d'une violation de la symetrie par rapport de renversement de temps (TRS) dans les series perturbatives. La solution de ce probleme amene a une rederivation correcte de MCT et donne des series perturbatives respectantes TRS. Cette construction permet de verifier directement la stabilite structurelle de MCT ce qui permet de repondre a une question principale de la theorie: Est ce que la transition est coupee par dans le cadre d'une approximation plus astucieuse ou il y a des raisons plus fondamentales ? Le reponse est que MCT n'est qu'une theorie de champ moyen et, par consequence, les corrections pertrubatives sont irrelevantes. Dans le meme contexte on etudie aussi le mapping des liquides surfondus sur les systemes de reaction-diffusion. Les resultats donnes au-dessus sont valables au-dessus de la transition vitreuse i.e pour les liquides en equilibre. Dans la derniere chapitre on etudie la generalisation de ces resultats aux basses temperatures ou le systeme n'atteint jamais l'equilibre pendant l'observation et le temps relaxation est, effectivement, infini. L'analyse est plus complique dans ce cas car des nouveaux phenomene, comme le vieillissement des proprietes physique, apparaissent