Recovering the equivalence of statistical ensembles

The equivalence of thermodynamic results in the canonical and the microcanonical ensembles has been questioned in a number of calculations for lattice spin models with equivalent-neighbor (mean-field) interactions. We show that these claims of inequivalence are related to an inadequate definition of the independent (density) variables in the microcanonical ensemble. We illustrate this point by revisiting the paradigmatic examples given by Ruffo, Mukamel, and coworkers: (i) A fully-connected spin-1 Blume-Capel model, which is known to display a phase diagram with a tricritical point; (ii) An Ising chain with competing short and long-range interactions, which displays second and first-order phase transitions for suitable choices of parameters (see A. Campa, T. Dauxois, and S. Ruffo, Phys. Repts. 480, 57-159, 2009). Although long-range interactions pose notoriously difficult problems, we show that in these elegant and paradigmatic cases there is no disagreement between calculations in the usual canonical and microcanonical ensembles. The crucial point is the recognition that the microcanonical ensemble describes an isolated system, in terms of suitable thermodynamic density variables, while the usual canonical ensemble is characterized by the thermodynamically conjugate field variables. Details of our arguments are described in V. B. Henriques and S. R. Salinas, Recovering the equivalence of ensembles, arxiv: 1501.04029v1 (and in part II, to be posted soon).