Recovering the equivalence of ensembles II: An Ising chain with competing short and long-range interactions

TL;DR

This paper revisits Nagle's Ising chain model with competing interactions, demonstrating the equivalence of canonical and microcanonical ensembles under proper variable choices, and clarifying previous claims of inequivalence.

Contribution

It provides a detailed analysis showing ensemble equivalence in Nagle's model, correcting recent misconceptions about microcanonical and canonical differences.

Findings

Canonical and microcanonical results are equivalent with proper variables

The model exhibits both second and first-order phase transitions

Ensemble inequivalence claims are refuted with correct formulation

Abstract

In a pioneer work, John Nagle has shown that an Ising chain with competing short and long-range interactions displays second and first-order phase transitions separated by a tricritical point. More recently, it has been claimed that Nagle's model provides an example of the inequivalence between canonical and microcanonical calculations. We then revisit Nagle's original solution, as well as the usual formulation of the problem in a canonical ensemble, which lead to the same results. Also, in contrast to recent claims, we show that an alternative formulation in the microcanonical ensemble, with the adequate choice of the fixed thermodynamic extensive variables, leads to equivalent thermodynamic results.