Understanding the macroscopic properties of matter, based on the microscopic interactions of the single particles requires to bring together the areas of statistical physics and dynamical systems. For deterministic diffusion one of the most prominent models is the Lorentz gas in which a point particle performs specular reflections with hard disks distributed in the plane. This model generates deterministic chaos and has led to many mathematical results revealing the origin of diffusion starting from chaotic dynamics. For the periodic Lorentz gas on a triangular lattice, it is possible to understand the diffusion coefficient, in the limit of high scatterer densities, in terms of random walk approximations. The key question addressed in this thesis is: What happens to the diffusion coefficient, as a function of control parameters, if the hard potential walls of the Lorentz gas scatterers are replaced by a soft potential? In this study we use a repulsive Fermi potential from which the hard limit can be recovered by varying a control parameter. We then performed computer simulations and analytical random walk approximations to understand the functional form of the diffusion coefficient as a function of the following parameters: the minimal distance between two scatters, the softness of the potential and the energy of a moving particle. Our main results is that the diffusion coefficient is a highly irregular function of each of these control parameters. Under certain assumptions one can construct analytical approximations that describe the coarse shape of the diffusion coeffi- cient when it exists: For high densities of scatterers we develop suitable random walk approximations, in the low density regime we apply a more elaborate argument that tests for memory effects. We find that diffusion in our soft Lorentz gas exhibits different random walk regimes, where either randomization characterizes the evolution of diffusion or spatio-temporal correlations take place. Via Poincare surfaces of section we show that the irregularities appearing in the diffusion coef- cient, as a function of parameters, which strongly deviate from simple random walk dynamics, come from non-trivial quasi-ballistic trajectories in con guration space.

An introduction to periodic orbit techniques for deterministic dynamical systems is presented. The Farey map is considered as examples of intermittency in one-dimensional maps. The effect of intermittency on the Markov partition is considered. The Gauss map is shown to be related to the farey map by a simple transformation of trajectories. A method of calculating periodic orbits in the thermostated Lorentz gas is derived. This method relies on minimising the action from the Hamiltonian description of the Lorentz gas, as well as the construction of a generating partition of the phase space. This method is employed to examine a range of bifurcation processes in the Lorentz gas. A novel construction of the Sinai billiard is performed by using symmetry arguments to reduce two particles in a hard walled box to the square Sinai billiard. Infinite families of periodic orbits are found, even at the lowest order, due to the intermittency of the system. The contribution of these orbits is examined and found to be tractable at the lowest order. The number of orbits grows too quickly for consideration of any other terms in the periodic orbit expansion. A simple stochastic model for the diffusion in the Lorentz gas was constructed. The model produced a diffusion coefficient that was a remarkably good fit to more precise numerical calculations. This is a significant improvement to the Machta-Zwanzig approximation for the diffusion coefficient. We outline a general approach to constructing stochastic models of deterministic dynamical systems. This method should allow for calculations to be performed in more complicated systems.

This thesis covers my work in the field of closed, repeatedly driven, Hamiltonian systems. These systems do not exchange particles with the surrounding environment and their time-evolution is described by Hamilton's equations of motion (in the classical framework) or the Schroedinger equation (in the quantum framework). Their interaction with the environment is encoded into the time-dependence of the system's Hamiltonian. Chapter 1 is an "Overview" in which the status of the field, my contributions and future prospective are outlined. Chapters 2 to 4 provide the theoretical background which is used in Chapters 5 to 7 to derive some original results. These results show that in Hamiltonian systems, after many driving events, universal properties emerge. In particular, using the framework of the linear Boltzmann equation, I have studied the dynamics of a mobile, light impurity in a gas of heavy particles. The impurity's kinetic energy increases and, in the long time limit, approaches a non-thermal asymptotic distribution. The significance of this work is to show explicitly the emergence of a non-thermal distribution in a closed, driven system. Moreover, using the work-fluctuation theorems, I have studied the character of the energy distribution of a generic isolated system driven according a generic protocol. Both thermal and non-thermal distributions can be realized for the same system by changing the characteristics of the driving protocol. These two different regimes are separated by a dynamical phase transition. Finally, I have used the Floquet Theory and the Magnus Expansion to analyze the behavior of a generic interacting system which is driven periodically in time. For fast driving the system is unable to absorb energy and remains localized in the low energy part of the Hilbert space while for slow driving the system absorbs energy and, in the long time limit, it is delocalized in the entire Hilbert space. These two qualitatively different behaviors are separated by a many-body localization transition which is related to the break down of the Magnus expansion at the critical value of the driving frequency.

Dans cette thèse, nous étudions la généralisation à une infinité de particules d'un modèle hamiltonien décrivant les interactions entre une particule et son environnement. Le milieu est considéré comme une superposition continue de membranes vibrantes. Au bout d'un certain temps, tout se passe comme si la particule était soumise à une force de frottement linéaire. Les équations obtenus pour un grand nombre de particules sont proches des équations de Vlasov. Dans un premier chapitre, on montre d'abord l'existence et l'unicité des solutions puis on s'intéresse à certains régimes asymptotiques; en faisant tendre la vitesse des ondes dans le milieu vers l'infini et en redimensionnant les échelles, on obtient à la limite une équation de Vlasov, on montre que si l'on modifie en plus une fonction paramètrisant le système, on obtient l'équation de Vlasov-Poisson attractive. Dans un deuxième chapitre, on ajoute un terme de diffusion à l'équation. Cela correspond à prendre en compte une agitation brownienne et un frottement linéaire sur les particules. Le principal résultat de ce chapitre est la convergence de la distribution de particules vers une unique distribution stationnaire. On montre la limite de diffusion pour ce nouveau système en faisant tendre simultanément la vitesse de propagation vers l'infini. On obtient une équation plus simple pour la densité spatiale. Dans le chapitre 3, nous montrons la validité des équations déjà étudiées par une limite de champ moyen. Dans le dernier chapitre, on étudie l'asymptotique en temps long de l'équation décrivant l'évolution de la densité spatiale obtenue dans le chapitre 2, des résultats faibles de convergence sont obtenus / The goal of this PhD is to study a generalisation of a model describing the interaction between a single particle and its environment. We consider an infinite number of particles represented by their distribution function. The environment is modelled by a vibrating scalar field which exchanges energy with the particles. In the single particle case, after a large time, the particle behaves as if it were subjected to a linear friction force driven by the environment. The equations that we obtain for a large number of particles are close to the Vlasov equation. In the first chapter, we prove that our new system has a unique solution. We then care about some asymptotic issues; if the wave velocity in the medium goes to infinity, adapting the scaling of the interaction, we connect our system with the Vlasov equation. Changing also continuously a function that parametrizes the model, we also connect our model with the attractive Vlasov-Poisson equation. In the second chapter, we add a diffusive term in our equation. It means that we consider that the particles are subjected to a friction force and a Brownian motion. Our main result states that the distribution function converges to the unique equilibrium distribution of the system. We also establish the diffusive limit making the wave velocity go to infinity at the same time. We find a simpler equation satisfied by the spatial density. In chapter 3, we prove the validity of both equations studied in the two first chapters by a mean field limit. The last chapter is devoted to studying the large time asymptotic properties of the equation that we obtained on the spatial density in chapter 2. We prove some weak convergence results