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

## TL;DR

This paper critiques a longstanding derivation linking phenomenological Tsallis distributions to Tsallis statistics, showing that previous results were based on inconsistent definitions of expectation values, leading to different distribution functions.

## Contribution

It clarifies the inconsistency in prior derivations by using expectation values consistent with Tsallis-2 statistics, correcting the link between phenomenological distributions and Tsallis statistics.

## Key findings

- Previous derivations used inconsistent expectation value definitions.
- Corrected distributions differ from phenomenological Tsallis distributions.
- Establishes the importance of consistent expectation value definitions in Tsallis statistics.

## Abstract

From the Tsallis unnormalized (or Tsallis-2) statistical mechanical formulation, B\"{u}y\"{u}kkili\c{c} {\it et al.} [Phys. Lett. A 197, 209 (1995)] 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 using the definition of the generalized expectation values (of the Tsallis-2 statistics), but the single-particle distribution function was calculated from this probability distribution using the standard definition of the expectation values of the Tsallis normalized (or Tsallis-1) statistics. Considering the definition of the expectation values which is consistent with the Tsallis-2 formulation, we have proved that the single-particle (classical and quantum) distribution functions in the factorization approximation differ from the phenomenological Tsallis distributions.