Abstract
In the present paper the conditions for the validity of the Tsallis statistics are analyzed. The same has been done following the analogy with the traditional case: starting from the microcanonical description of the systems and taking into account their self-similarity scaling properties in the thermodynamic limit, it is analyzed the necessary conditions for the equivalence of microcanonical ensemble with the Tsallis generalization of the canonical ensemble. It is shown that the Tsallis statistics is appropriate for the macroscopic description of systems with potential scaling laws of the asymptotic accessible states density of the microcanonical ensemble. Our analysis shows many details of the Tsallis's formalism: the q-expectation values, the generalized Legendre transformations between the thermodynamic potentials, as well as the conditions for its validity, having a priori the possibility to estimate the value of the entropic index without the necessity of appealing to the computational simulations or the experiment. On the other hand, the definition of physical temperature received a modification that differs from the Toral result. For the case of finite systems, we have generalized the microcanonical thermostatistics of Gross with the generalization of the curvature tensor for this kind of description.
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