The notion of temperature in non-extensive systems

Temperature is a well defined concept in equilibrium Statistical Mechanics, however, its extension to non-extensive systems whose distribution of microstates belong to the q-exponential family is a controversial topic (see for example M. Nauenberg, Phys. Rev. E 67, 036114, 2003). In this work we discuss the implications of a recently derived identity (S. Davis and G. Gutierrez, Phys. Rev. E 86, 051136, 2012) for the estimation of the parameters $\beta$ and $q$ of the q-exponential analog of the canonical ensemble, \begin{equation} P(r, p) \propto \Theta(1-(1-q)\beta H)(1-(1-q)\beta H). \end{equation} We show that the expectation of the Rugh estimator \begin{displaymath} R(x, p) = \nabla \cdot \frac{\vec \omega}{\vec \omega \cdot \nabla H}, \end{displaymath} where $\vec \omega=\vec \omega(\vec r, \vec p)$ is an arbitrary differentiable field, plays the role of the inverse temperature of the system regardless of the statistical ensemble.