Dans cette thèse, nous étudions le comportement asymptotique précis à basse température de l’événement de sortie d'un domaine métastable $Omegasubset mathbb R^d$ (point de sortie et temps de sortie) pour le processus de Langevin sur amorti. En pratique, le processus de Langevin sur amorti peut par exemple simuler l'évolution des positions des atomes d'une molécule ou la diffusion d'impuretés interstitielles dans un cristal. Nos résultats principaux concernent le comportement asymptotique précis de la distribution de la loi du point de sortie de $Omega$. Dans la limite d'une petite température, ces résultats permettent de justifier l'utilisation de la formule d'Eyring-Kramers pour modéliser les événements de sortie de $Omega$. La loi d'Eyring-Kramers est par exemple utilisée pour calculer les taux de transition entre les états d'un système dans un algorithme de Monte-Carlo cinétique afin de simuler efficacement les différents états visités par le système. L'analyse repose de manière essentielle sur la distribution quasi stationnaire associée au processus de Langevin sur amorti dans $Omega$. Nos preuves utilisent des outils d'analyse semi-classique. La thèse se décompose en trois chapitres indépendants. Le premier chapitre (rédigé en français) est une introduction aux résultats obtenus. Les deux autres chapitres (rédigées en anglais) sont consacrés aux énoncés mathématiques / This thesis is dedicated to the study of the sharp asymptotic behaviour in the low temperature regime of the exit event from a metastable domain $Omegasubset mathbb R^d$ (exit point and exit time) for the overdamped Langevin process. In practice, the overdamped Langevin dynamics can be used to describe for example the motion of the atoms of a molecule or the diffusion of interstitial impurities in a crystal. The obtention of sharp asymptotic approximations of the first exit point density in the small temperature regime is the main result of this thesis. These results justify the use of the Eyring-Kramers law to model the exit event. The Eyring-Kramers law is used for example to compute the transition rates between the states of a system in a kinetic Monte-Carlo algorithm in order to sample efficiently the state-to-state dynamics. The cornerstone of our analysis is the quasi stationary distribution associated with the overdamped Langevin dynamics in $Omega$. The proofs are based on tools from semi-classical analysis. This thesis is divided into three independent chapters. The first chapter (in French) is dedicated to an introduction to the mathematical results. The other two chapters (in English) are devoted to the precise statements and proofs

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

This thesis describes an experimental study on fluctuations of a Brownian particle immersed in a fluid, confined by optical tweezers and subject to two different kinds of non-equilibrium conditions. We aim to gain a rather general understanding of the relation between spontaneous fluctuations, linear response and total entropy production for processes away from thermal equilibrium. The first part addresses the motion of a colloidal particle driven into a periodic non-equilibrium steady state by a nonconservative force and its response to an external perturbation. The dynamics of the system is analyzed in the context of several generalized fluctuation-dissipation relations derived from different theoretical approaches. We show that, when taking into account the role of currents due to the broken detailed balance, the theoretical relations are verified by the experimental data. The second part deals with fluctuations and response of a Brownian particle in two different aging baths relaxing towards thermal equilibrium: a Laponite colloidal glass and an aqueous gelatin solution. The experimental results show that heat fluxes from the particle to the bath during the relaxation process play the same role of steady state currents as a non-equilibrium correction of the fluctuation-dissipation theorem. Then, the present thesis provides evidence that the total entropy production constitutes a unifying concept which links the statistical properties of fluctuations and the linear response function for non-equilibrium systems either in stationary or non stationary states.

Ces travaux de thèse présentent une étude expérimentale des fluctuations d'une particule Brownienne soumise à deux différentes conditions hors-équilibre dans un fluide . Le but est de comprendre d'une manière générale la relation entre les fluctuations spontanées, la fonction de réponse linéaire et la production totale d'entropie des processus loin de l'équilibre thermique. La première partie est consacrée à l'étude du mouvement d'une particule colloïdale dans un état stationnaire périodique hors-équilibre induit par une force non-conservative et à sa réponse à une perturbation externe. Nous analysons la dynamique du système dans le contexte des différentes approches généralisées de fluctuation-dissipation. Nous montrons que ces relations théoriques sont satisfaites par les données expérimentales quand on prend en compte le rôle du courant du à la rupture du bilan détaillé. Dans une deuxième partie nous étudions les fluctuations et la réponse d'une particule Brownienne dans deux types de bains vieillissants qui relaxent vers l'équilibre thermique: un verre colloïdal de Laponite et une solution aqueuse de gélatine. Dans ce cas-là nous montrons que le flux de chaleur de la particule vers le bain pendant sa relaxation représente une correction hors-équilibre du théorème de fluctuation-dissipation. Donc, le flux de chaleur joue le même rôle que le courant dans un état stationnaire. En conséquence, les résultats de la thèse mettent en évidence l'importance générale de la production totale d'entropie pour quantifier les relations de fluctuation-dissipation généralisées dans les systèmes hors-équilibre. / This thesis describes an experimental study on fluctuations of a Brownian particle immersed in a fluid, confined by optical tweezers and subject to two different kinds of non-equilibrium conditions. We aim to gain a rather general understanding of the relation between spontaneous fluctuations, linear response and total entropy production for processes away from thermal equilibrium. The first part addresses the motion of a colloidal particle driven into a periodic non-equilibrium steady state by a nonconservative force and its response to an external perturbation. The dynamics of the system is analyzed in the context of several generalized fluctuation-dissipation relations derived from different theoretical approaches. We show that, when taking into account the role of currents due to the broken detailed balance, the theoretical relations are verified by the experimental data. The second part deals with fluctuations and response of a Brownian particle in two different aging baths relaxing towards thermal equilibrium: a Laponite colloidal glass and an aqueous gelatin solution. The experimental results show that heat fluxes from the particle to the bath during the relaxation process play the same role of steady state currents as a non-equilibrium correction of the fluctuation-dissipation theorem. Then, the present thesis provides evidence that the total entropy production constitutes a unifying concept which links the statistical properties of fluctuations and the linear response function for non-equilibrium systems either in stationary or non stationary states.

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.