Long-time and large-scale behaviors of some collision dynamics.

Summary This thesis consists of three essentially independent parts, each of which is devoted to the study of a system of particles, following a deterministic or random dynamics, and inside which the interactions are done only at the collisions between the particles.Part I proposes a numerical and theoretical study of the non-equilibrium stationary states of the Complete Exchange Model, introduced in physics to understand the transport of heat in some porous materials.Part II is devoted to a system of Brownian particles evolving on the real straight line and interacting through their rank. The limiting behavior of this system, in long time and with a large number of particles, is described, then the results are applied to the study of a financial market model called mean field Atlas model.Part III introduces a multitype version of the particle system studied in the previous part, which allows to approach parabolic systems of nonlinear partial differential equations. The small noise limit of this system is called multitype sticky particle dynamics and approaches this time hyperbolic systems. A detailed study of this dynamics gives stability estimates in Wasserstein distance on the solutions of these systems.