ar X iv : c on d - m at / 0 40 86 81 v 1 3 1 A ug 2 00 4 The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

Microcanonical equilibrium macrostates are characterized as the solutions of a constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions of a related, unconstrained minimization problem. In the paper [19] by Ellis, Haven, and Turkington, the problem of ensemble equivalence was completely solved at two separate, but related levels: the level of equilibrium macrostates, which focuses on relationships between the corresponding sets of equilibrium macrostates, and the thermodynamic level, which focuses on when the microcanonical entropy s can be expressed as the Legendre-Fenchel transform of the canonical free energy. A neat but not quite precise statement of the main result in [19] is that the microcanonical and 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 the microcanonical entropy s is concave. The present paper extends the results in [19] 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 that we consider 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 [19], 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 present paper 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 the generalized microcanonical entropy s − g is concave. 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. For example, if the microcanonical entropy is C, then universal equivalence of ensembles holds with g taken from a class of quadratic functions. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization.