Many natural systems are far from thermodynamic equilibrium and keep on exchanging matter, energy or information with their surroundings. These exchanges produce currents, or fluxes, that break time-reversal invariance. Such systems lie beyond the realm of traditional thermodynamics and the principles of equilibrium statistical mechanics do not apply to them. In fact, there exists no general conceptual framework a la Gibbs-Boltzmann to describe these systems from first principles.
The last two decades, however, have witnessed remarkable progress. The aim of this lecture is to explain some recent developments, such as the Work Identities, the Fluctuation Theorem (Cohen, Evans, Gallavotti and Morriss) and the Macroscopic Fluctuation Theory which represent the first steps towards a unified approach to non-equilibrium behaviour. These concepts will be illustrated with some simple systems such as the Brownian ratchet model for molecular motors and the asymmetric exclusion process, which is considered today as the paradigm of non-equilibrium physics.