Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems

TL;DR

This paper investigates when the mean-field theory accurately describes long-range interacting spin systems in the microcanonical ensemble, revealing conditions where it fails due to ensemble inequivalence.

Contribution

It derives necessary and sufficient conditions for the exactness of mean-field theory in nonadditive long-range systems and applies them to specific Potts models.

Findings

Mean-field theory may not be exact if microcanonical and canonical ensembles differ.

Conditions for mean-field exactness depend on ensemble equivalence.

Application to -Potts models illustrates these conditions.

Abstract

Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called "exactness of the mean-field theory". It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the \alpha-Potts model with annealed vacancies and the \alpha-Potts model with invisible states.