An account of the statistical and dynamical properties of systems with long-range interactions

We shortly review recent progress in the field of long-range interactions, analyzed from the viewpoint of statistical mechanics and of dynamical systems theory. Long-range interaction potentials decay at large distance with a power that is smaller than space dimension. This feature causes a lack of additivity, which in turn implies inequivalence between microcanonical and canonical ensemble for what concerns predictions near first order canonical phase transitions. Inequivalence manifests itself physically by the presence of curious effects such as negative specific heat and temperature jumps. We also briefly mention dynamical effects which we believe are generic for long-range interactions: (i) quasi-stationary states whose lifetime increases with system size; (ii) metastable states that live proportionally to the exponential of systems size times the entropy barrier; (iii) broken ergodicity.