Superstatistics and the quest of generalized ensembles equivalence in a system with long-range interactions

TL;DR

This paper uses superstatistics to analyze the Blume-Capel model, demonstrating ensemble equivalence and unifying canonical and microcanonical descriptions through a nonextensive parameter.

Contribution

It introduces a superstatistics approach to connect canonical and microcanonical ensembles in long-range interacting systems, unifying previous generalized ensemble results.

Findings

Superstatistics recovers both canonical and microcanonical solutions.

EGE and BC solutions are thermodynamically equivalent.

The nonextensive parameter smoothly interpolates between ensembles.

Abstract

The so-called $\chi^{2}$-superstatistics of Beck and Cohen (BC) is employed to investigate the infinite-range Blume-Capel model, a well-known representative system displaying inequivalence of canonical and microcanonical phase diagrams. While not being restricted to any of those particular thermodynamic limits, our analytical result can smoothly recover both canonical and microcanonical ensemble solutions as its nonextensive parameter $q$ is properly tuned. Additionally, we compare our findings to ones previously obtained from a generalized canonical framework named Extended Gaussian ensemble (EGE). Finally, we show that both EGE and BC solutions are equivalent at the thermodynamic level.