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

TL;DR

This paper investigates the invariance properties of various nonextensive statistical formalisms under energy spectrum translation, confirming invariance for some and identifying limitations for others, thus clarifying their consistency with equilibrium statistical mechanics.

Contribution

It rigorously proves the invariance of Tsallis-1 and Tsallis-3 statistics under energy translation, and highlights the non-invariance of Tsallis-2, clarifying their physical consistency.

Findings

Tsallis-1 and Tsallis-3 distributions are invariant under energy translation.

Tsallis-2 distribution is not invariant under energy translation.

Boltzmann-Gibbs and q-dual statistics are invariant under energy translation.

Abstract

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.