Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model

TL;DR

This paper introduces a new way to characterize nonequivalent statistical ensembles using the mean-field BEG model, highlighting differences at the level of equilibrium distributions and extending thermodynamic criteria.

Contribution

It provides a novel statistical characterization of ensemble nonequivalence based on equilibrium distributions, complementing existing thermodynamic approaches.

Findings

Microcanonical distributions are not realized in the canonical ensemble where entropy is nonconcave.

Concave entropy regions show a one-to-one correspondence between microcanonical and canonical distributions.

Numerical results support theoretical links between thermodynamic and statistical ensemble equivalence.

Abstract

We illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume-Emery-Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy. By computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the BEG model, we show that for values of the mean energy where the microcanonical entropy is nonconcave, the microcanonical distributions of states are nowhere realized in the canonical ensemble. Moreover, we show that for values of the mean energy where the microcanonical entropy is strictly concave, the equilibrium microcanonical distributions of states can be put in one-to-one correspondence with equivalent canonical equilibrium distributions of states. Our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by Ellis, Haven, and Turkington [J. Stat. Phys. 101, 999 (2000)].