Some recent developments in non-equilibrium statistical physics

We first recall the laws of classical thermodynamics and the fundamental principles of statistical mechanics and emphasize the fact that the fluctuations of a system in macroscopic equilibrium, such as Brownian motion, can be explained by statistical mechanics and not by thermodynamics. In the vicinity of equilibrium, the susceptibility of a system to an infinitesimal external perturbation is related to the amplitude of the fluctuations at equilibrium (Einstein's relation) and exhibits a symmetry discovered by Onsager. We shall then focus on the mathematical description of systems out of equilibrium using Markovian dynamics. This will allow us to present some remarkable relations derived during the last decade and valid arbitrarily far from equilibrium: the Gallavotti-Cohen fluctuation theorem and Jarzynski's non-equilibrium work identities. These recent results will be illustrated by applying them to simple systems such as the Brownian ratchet model for molecular motors and the asymmetric exclusion process which is a basic example of a driven lattice gas.