Extended gaussian ensemble solution and tricritical points of a system with long-range interactions

TL;DR

This paper analyzes the extended gaussian ensemble (EGE) for the Blume-Capel model with long-range interactions, revealing how microcanonical states are recovered and ergodicity is broken without taking the limit of infinite gaussian parameter.

Contribution

It provides an explicit analytical solution of the EGE for the Blume-Capel model and demonstrates the recovery of microcanonical states and ergodicity breaking without the infinite limit.

Findings

Microcanonical states are recovered at finite gaussian parameter values.

Ergodicity breaking occurs due to inaccessible magnetic states at low energies.

The EGE solution clarifies differences between microcanonical and canonical phase transitions.

Abstract

The gaussian ensemble and its extended version theoretically play the important role of interpolating ensembles between the microcanonical and the canonical ensembles. Here, the thermodynamic properties yielded by the extended gaussian ensemble (EGE) for the Blume-Capel (BC) model with infinite-range interactions are analyzed. This model presents different predictions for the first-order phase transition line according to the microcanonical and canonical ensembles. From the EGE approach, we explicitly work out the analytical microcanonical solution. Moreover, the general EGE solution allows one to illustrate in details how the stable microcanonical states are continuously recovered as the gaussian parameter $\gamma$ is increased. We found out that it is not necessary to take the theoretically expected limit $\gamma \to \infty$ to recover the microcanonical states in the region between the canonical and microcanonical tricritical points of the phase diagram. By analyzing the entropy as a function of the magnetization we realize the existence of unaccessible magnetic states as the energy is lowered, leading to a treaking of ergodicity.