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The general formalisms of the $q$-dual statistics, the Boltzmann-Gibbsstatistics, and three versions of the Tsallis statistics known as Tsallis-1,Tsallis-2, and Tsallis-3 statistics have been considered in the canonicalensemble. We have rigorously proved that the probability distribution of theTsallis-1 statistics is invariant under the uniform energy spectrum translationat a fixed temperature. This invariance demonstrates that the formalism of theTsallis-1 statistics is consistent with the fundamentals of the equilibriumstatistical mechanics. The same results we have obtained for the probabilitydistributions of the Tsallis-3 statistics, Boltzmann-Gibbs statistics, and$q$-dual statistics. However, we have found that the probability distributionof the Tsallis-2 statistics, the expectation values of which are not consistentwith the normalization condition of probabilities, is indeed not invariantunder the overall shift in energy as expected.

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