Partial equivalence of statistical ensembles in a simple spin model with discontinuous phase transitions

TL;DR

This paper investigates phase transitions in a simple spin model, revealing how different statistical ensembles exhibit partial equivalence and distinct fluctuation behaviors during discontinuous transitions.

Contribution

It demonstrates the partial equivalence of microcanonical, canonical, and grand canonical ensembles in a spin model with discontinuous phase transitions, highlighting differences in fluctuation patterns.

Findings

Discontinuous transitions occur in all ensembles but differ in fluctuation behavior.

Microcanonical ensemble shows no fluctuations at transition.

Grand canonical ensemble exhibits mixed-order transition with diverging fluctuations.

Abstract

In this paper, we draw attention to the problem of phase transitions in systems with locally affine microcanonical entropy, in which partial equivalence of (microcanonical and canonical) ensembles is observed. We focus on a very simple spin model, that was shown to be an equilibrium statistical mechanics representation of the biased random walk. The model exhibits interesting discontinuous phase transitions that are simultaneously observed in the microcanonical, canonical, and grand canonical ensemble, although in each of these ensembles the transition occurs in a slightly different way. The differences are related to fluctuations accompanying the discontinuous change of the number of positive spins. In the microcanonical ensemble, there is no fluctuation at all. In the canonical ensemble, one observes power-law fluctuations, which are, however, size-dependent and disappear in the thermodynamic limit. Finally, in the grand canonical ensemble, the discontinuous transition is of mixed-order (hybrid) kind with diverging (critical-like) fluctuations. In general, this paper consists of many small results, which together make up an interesting example of phase transitions that are not covered by the known classifications of these phenomena.