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Ensemble equivalence and phase transitions for general models in statistical mechanics and for the Curie -Weiss -Potts model
Abstract
Using the theory of large deviations, we analyze in the first chapter the phase transition structure of the Curie-Weiss-Potts spin model, which is a mean-field approximation to the Potts model. This analysis is carried out both for the canonical ensemble and the microcanonical ensemble. Besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles, we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates, relating these to concavity and support properties of the microcanonical entropy. In the second chapter we extend the results in [23] significantly by addressing the following motivational question. Given that the microcanonical ensemble is not equivalent with the canonical ensemble, is it possible to replace the canonical ensemble with a generalized canonical ensemble that is equivalent with the microcanonical ensemble? The generalized canonical ensemble is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. The special case in which g is quadratic plays a central role in the theory, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. As in [23], we analyze the equivalence of the two ensembles at both the level of equilibrium macrostates and the thermodynamic level. A neat but not quite precise statement of the main result in the second chapter is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if s - g is concave. In order to carry out the analysis of ensemble equivalence, two forms of a generalized Legendre-Fenchel transform involving g are introduced and their properties are studied. The considerable freedom that one has in choosing g has the important consequence that even when the microcanonical and standard canonical ensembles are not equivalent, one can often find g with the property that the microcanonical and generalized canonical ensembles satisfy a strong form of equivalence which we call universal equivalence.
Subject Area
Mathematics
Recommended Citation
Costeniuc, Marius F, "Ensemble equivalence and phase transitions for general models in statistical mechanics and for the Curie -Weiss -Potts model" (2005). Doctoral Dissertations Available from Proquest. AAI3193892.
https://scholarworks.umass.edu/dissertations/AAI3193892