Recovering the equivalence of ensembles

TL;DR

This paper demonstrates that apparent inequivalence between the canonical and microcanonical ensembles in spin models with long-range interactions arises from improper variable definitions, and shows they are equivalent when correctly formulated.

Contribution

It clarifies the conditions under which the ensembles are equivalent by emphasizing proper variable definitions, resolving previous apparent discrepancies.

Findings

No fundamental inequivalence when variables are correctly defined.

Illustration with spin-1 ideal paramagnet and Blume-Capel model.

Reconciliation of microcanonical and canonical results.

Abstract

The equivalence of thermodynamic results in the canonical and the microcanonical ensembles has been questioned in some calculations for spin models with long-range 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 with the example of a simple spin-1 ideal paramagnet, and then revisit the original calculations of Barr\'e, Mukamel, and Ruffo, for a mean-field spin-1 Blume-Capel model. If the microcanonical ensemble is defined in terms of adequate density variables, we show that there is no disagreement with the calculations in the usual canonical ensemble (with fixed thermodynamic field variables).