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In the present work, we have found that the phenomenological Tsallisdistribution (which nowadays is largely used to describe the transversemomentum distributions of hadrons measured in $pp$ collisions at high energies)is consistent with the basis of the statistical mechanics if it belongs to the$q$-dual nonextensive statistics instead of the Tsallis one. We have definedthe $q$-dual statistics based on the $q$-dual entropy which was obtained fromthe Tsallis entropy under the multiplicative transformation of the entropicparameter $q\to 1/q$. We have found that the phenomenological Tsallisdistribution is equivalent to the transverse momentum distribution of the$q$-dual statistics in the zeroth term approximation. Since the $q$-dualstatistics is properly defined, it provides a correct link between thephenomenological Tsallis distribution and the second law of thermodynamics.

Equivalence of the phenomenological Tsallis distribution to the transverse momentum distribution of $q$-dual statistics