  In the present work, we have found that the phenomenological Tsallis
distribution (which nowadays is largely used to describe the transverse
momentum 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 defined
the $q$-dual statistics based on the $q$-dual entropy which was obtained from
the Tsallis entropy under the multiplicative transformation of the entropic
parameter $q\to 1/q$. We have found that the phenomenological Tsallis
distribution is equivalent to the transverse momentum distribution of the
$q$-dual statistics in the zeroth term approximation. Since the $q$-dual
statistics is properly defined, it provides a correct link between the
phenomenological Tsallis distribution and the second law of thermodynamics.
