Remarks on the phenomenological Tsallis distributions and their link with the Tsallis statistics

From the Tsallis unnormalized (or Tsallis-2) statistical mechanical formulation, Buyukkilic et al (1995 Phys. Lett. A 197, 209) derived the expressions for the single-particle distribution functions (known as the phenomenological Tsallis distributions) for particles obeying the Maxwell-Boltzmann, Bose-Einstein and the Fermi-Dirac statistics using the factorization approximation. In spite of the fact that this paper was published long time ago, its results are still extensively used in many fields of physics, and it is considered that it was this paper that established the connection between the phenomenological Tsallis distributions and the Tsallis statistics. Here we show that this result is incorrect: the mistake lies in the fact that the probability distribution function was derived from the maximum entropy principle using the definition of the generalized expectation values (of the Tsallis-2 statistics), but the single-particle distribution functions were calculated on the basis of this probability distribution of the Tsallis-2 statistics using the standard definition of the expectation values of the Tsallis normalized (or Tsallis-1) statistics. Considering the definition of the expectation values for thermodynamic quantities which is consistent with the definition of expectation values of the constraints in the maximum entropy principle of the Tsallis-2 formulation, we have proved that the single-particle (classical and quantum) distribution functions in the factorization approximation differ from the well-known phenomenological Tsallis distributions.