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

Superstatistics inception by Beck and Cohen was intended to provide
an extension of the standard statistical mechanics formalism into
a more general one, focusing on describing out-of-equilibrium systems,
which are most likely characterized by spatio-temporal fluctuations
of an intensive parameter. Its usual formulation employs, as a working
hypothesis, the argument that fluctuations evolve on a long-time scale,
while the studied system can still be locally decomposed in many small
cells (subsystems) obeying the (equilibrium) statistical mechanics
characterized, for instance, by an effective local (inverse) temperature
$\beta$. For such systems, not only the temperature environment is
considered to be a fluctuating quantity, with probability density
$f\left(\beta\right)$, but also it may carry a spatial modulation
as a classical scalar field.

Here, the thermodynamic properties yielded by the nonextensive Statistical
Mechanics of Tsallis are derived, as a particular limit from the Superstatistics
approach, for the Blume-Capel (BC) model with infinite-range interactions.
This model exhibits nonconcavity of the entropy as a function of the
energy. Thus, it is well-known to present ensemble inequivalence,
which implies on different predictions for the first-order phase transition
line when taking its microcanonical or canonical description. Starting
from the Superstatistics approach, we numerically investigate how
the microcanonical limit can be recovered as a function of the nonextensive
parameter $q$ and system-size $\left(N\right)$. Moreover, we highlight
how this solution can be compared to our previous work where BC model
was solved in an interpolating generalized ensemble, known as Extented
Gaussian Ensemble (EGE), which is able to continuously recover the
stable microcanonical states as its ``nonextensive'' parameter $\gamma$
is gradually increased. In this vein, we found out that it is not
necessary to take the theoretically expected limit $q\rightarrow1$
to recover the microcanonical states in the region between the canonical
and microcanonical tricritical points of the BC phase diagram.