Large Deviations Techniques for Long-Range Interactions

Abstract

AbstractAfter 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 Cramèr’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.KeywordsOrder Phase TransitionCanonical EnsembleFree Electron LaserTricritical PointMicrocanonical EnsembleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.