Thermodynamic Framework for a Nonextensive System

Recently, an effective temperature $\theta$
was introduced within the context of interacting vortices in type II
superconductors.
The quantity $\theta$ was shown to represent
an appropriate definition of effective temperature
for this system, exhibiting
properties very similar to those of the usual
thermodynamic temperature $T$, being:
(a) A positive quantity by definition;
(b) Thermodynamically conjugated to a
generalized entropy per particle, $s_{q}$ with $q=2$,
characteristic of nonextensive statistical mechanics.
In this way, a heat contribution was defined,
$\delta Q = \theta ds_{2}$;
(c) Proportional to the density of vortices $n$.
This property yields the desirable possibility for
varying $\theta$, since
recent experimental researches in type II superconductors
led to considerable advances in the ability of controlling
many properties of these vortices, including their
density;
(d) Characterized by values that are much higher than
typical room temperatures ($\theta \gg T$), so that
the thermal noise can be neglected as
a good approximation ($T/\theta \simeq 0$);
(e) Physically interpreted
in terms of the variance of the vortex positions,
$\theta \propto \langle x^{2} \rangle ^{3/2}$;
(f) Consistent with the definition of a Carnot
cycle, whose efficiency was shown to be
$\eta=1-(\theta_{2}/\theta_{1})$, where
$\theta_{1}$ and $\theta_{2}$ represent the effective temperatures
associated with the isothermal transformations of the cycle, with
$\theta_{1} > \theta_{2}$.
In order to achieve this later result,
an infinitesimal-work term $\delta W$ was introduced, leading
to a proposal for the first law of thermodynamics.
In the present work we explore the heat contribution,
$\delta Q = \theta ds_{2}$, by considering systems
in thermal contact in such a way to exchange heat among
themselves. Important concepts like thermal equilibrium
and heat reservoir are introduced, and particularly,
the zeroth principle is established.
Moreover, we consolidate the first-law proposal
by following the usual procedure for obtaining
different potentials, i.e., applying Legendre transformations
for distinct pairs of independent variables.
From these potentials we derive the equation of state,
Maxwell relations, and define response functions.
All results presented are shown to be consistent with those
of standard thermodynamics for $T>0$.