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

• Exact χ2χ2-superstatistics solution of infinite-range BEC model is provided.
• χ2χ2-superstatistics is shown to interpolate from canonical to microcanonical ensembles.
• χ2χ2-superstatistics is shown to be equivalent to EGE solution for BEC model.

The so-called χ2χ2-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 qq 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.