Large deviation techniques applied to systems with long-range interactions

We discuss a method to solve models with {\it long-range interactions} in the microcanonical and canonical ensemble. The method closely follows the one introduced by R.S. Ellis, Physica D 133, 106 (1999), which uses large deviation techniques. We show how it can be adapted to obtain the solution of a large class of simple models, which can show {\it ensemble inequivalence}. The model Hamiltonian can have both {\em discrete} (Ising, Potts) and {\em continuous} (HMF, Free Electron Laser) state variables. This latter extension gives access to the comparison with dynamics and to the study of non-equilibrium effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the $\alpha$-Ising model in one-dimension with $0\leq \alpha