  The analysis of high-energy particle collisions is an excellent testbed for
the non-extensive statistical approach. In these reactions we are far from the
thermodynamical limit. In small colliding systems, such as electron-positron or
nuclear collisions, the number of particles is several orders of magnitude
smaller than the Avogadro number; therefore, finite-size and fluctuation
effects strongly influence the final-state one-particle energy distributions.
Due to the simple characterization, the description of the identified hadron
spectra with the Boltzmann-Gibbs thermodynamical approach is insufficient.
These spectra can be described very well with Tsallis-Pareto distributions
instead, derived from non-extensive thermodynamics. Using the $q$-entropy
formula, we interpret the microscopic physics in terms of the Tsallis $q$ and
$T$ parameters. In this paper we give a view on these parameters, analyzing
identified hadron spectra from recent years in a wide center-of-mass energy
range. We demonstrate that the fitted Tsallis-parameters show dependency on the
center-of-mass energy and particle species (mass). Our findings are described
well by a QCD (Quantum Chromodynamics) inspired parton evolution ansatz. Based
on this comprehensive study, apart from the evolution, both mesonic and
baryonic components found to be non-extensive ($q>1$), besides the mass ordered
hierarchy observed in the parameter $T$. We also study and compare in details
the theory-obtained parameters for the case of PYTHIA8 Monte Carlo Generator,
perturbative QCD and quark coalescence models.
