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

TL;DR

This paper demonstrates that the phenomenological Tsallis distribution used in high-energy physics can be derived from the $q$-dual nonextensive statistical mechanics framework, establishing a thermodynamically consistent basis.

Contribution

It introduces the $q$-dual statistics and shows its equivalence to the Tsallis distribution in describing transverse momentum data, linking phenomenology with fundamental thermodynamics.

Findings

Tsallis distribution is consistent with $q$-dual statistics

$q$-dual entropy derived from Tsallis entropy via $q o 1/q$ transformation

Phenomenological distribution aligns with $q$-dual transverse momentum distribution

Abstract

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.