Statistical mechanics of systems with long range interactions

Many physical systems are governed by long range interactions, the main example being self-gravitating stars. Long range interaction implies a lack of additivity for the energy. As a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many claims, the statistical mechanics of such systems is a well understood subject. In this proceeding, we explain briefly the classical way to equilibrium and non equilibrium statistical mechanics, starting from first principles and emphasizing some new results. At equilibrium, we explain how the Boltzmann-Gibbs entropy can be proved to be the appropriate one, using large deviations tools. We explain the thermodynamics consequences of the lack of additivity, like the generic occurence of statistical ensemble inequivalence and negative specific heat. This well known behavior is not the only way inequivalence may occur, as emphasized by a recent new classification of phase transitions and ensemble inequivalence in systems with long range interaction. We note a number of generic situations that have not yet been observed in any physical systems. Out of equilibrium, we show that algebraic temporal correlations or anomalous diffusion may occur in these systems, and can be explained using usual statistical mechanics and kinetic theory