Study of invariance of nonextensive statistics under the uniform energy spectrum translation

The general formalisms of the q-dual statistics, the Boltzmann-Gibbs statistics, and three versions of the Tsallis statistics known as Tsallis-1, Tsallis-2, and Tsallis-3 statistics have been considered in the canonical ensemble. We have rigorously proved that the probability distribution of the Tsallis-1 statistics is invariant under the uniform energy spectrum translation at a fixed temperature. This invariance demonstrates that the formalism of the Tsallis-1 statistics is consistent with the fundamentals of the equilibrium statistical mechanics. The same results we have obtained for the probability distributions of the Tsallis-3 statistics, Boltzmann-Gibbs statistics, and q-dual statistics. However, we have found that the probability distribution of the Tsallis-2 statistics, the expectation values of which are not consistent with the normalization condition of probabilities, is indeed not invariant under the overall shift in energy as expected. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This study examines the probability distributions of different statistical methods in the canonical ensemble, showing that the distributions of Tsallis-1 and Tsallis-3 statistics are invariant under energy spectrum translations, consistent with the principles of equilibrium statistical mechanics. However, the probability distribution of Tsallis-2 statistics does not exhibit the expected invariance under overall energy shifts, indicating a deviation from the normalization condition of probabilities.