Large deviations techniques for long-range interactions

After a brief introduction to the main equilibrium features of long-range interacting systems (ensemble inequivalence, negative specific heat and susceptibility, broken ergodicity, etc.) and a recall of Cramer’s theorem, we discuss in this chapter a general method which allows us to compute microcanonical entropy for systems of the mean-field type. The method consists in expressing the Hamiltonian in terms of global variables and, then, in computing the phase-space volume by fixing a value for these variables: this is done by using large deviations. The calculation of entropy as a function of energy is, thus, reformulated as the solution of a variational problem. We show the power of the method by explicitly deriving the equilibrium thermodynamic properties of the three-state Potts model, the Blume-Capel model, an XY spin system, the ϕ 4 model and the Colson-Bonifacio model of the free electron laser. When short range interactions coexist with long-range ones, the method cannot be straightforwardly applied. We discuss an alternative variational method which allows us to solve the XY model with both mean-field and nearest neighbor interactions.