Abstract
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 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.
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Patelli, A., Ruffo, S. (2014). Large Deviations Techniques for Long-Range Interactions. In: Vulpiani, A., Cecconi, F., Cencini, M., Puglisi, A., Vergni, D. (eds) Large Deviations in Physics. Lecture Notes in Physics, vol 885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54251-0_7
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DOI: https://doi.org/10.1007/978-3-642-54251-0_7
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