K+/π+ horn: appropriate statistics for dynamical nonequilibrium nonextensive particle production

We suggest that the incapability of the thermal models in reproducing the horn-like structure of the Kaon-to-pion ratios measured at AGS, SPS, and RHIC energies and then confirmed in the Beam Energy Scan programs is likely emerged from an ad hoc utilization of inappropriate types of statistics, either extensive additive Boltzmann-Gibbs (BG) or the Tsallis-type of nonextensivity. In this regard, we emphasize that any suggested statistical approach for the dynamical and likely nonequilibrium nonextensive particle production, for example, Kaon and pion yields, should not necessarily remain valid over the whole range of the collision energies. We claim that generic (non)extensive statistics with two equivalence classes (c,d) is the most suitable statistical approach to replace all those assumptions and conjectures. Accordingly, the statistical ensemble itself becomes autonomously capable to determine its degree of extensivity or nonextensivity, where these are not necessarily limited to BG orTsallis. The generic approach covers the entire (c,d)-space, in which both extensive BG and nonextensive Tsallis statistics are characterized by distinct equivalence classes, (1,1) and (0,q), respectively. The energy dependence of the light-, γq, and strange-quark occupation factor, γs, indicates that the produced particles are best described as a nonequilibrium ensemble, where the remarkable nonmonotonic behavior associating the horn-structure of K+/π+ ratio is qualitatively reproduced. On other hand, the resulting equivalence classes (c,d) refer to a generic nonextensivity associated with an extended exponential and a Lambert-W0 exponentially generating distribution function, which obviously emerge from free, short- and long-range correlations over the wide range of the collision energies. With this generic statistical approach, the hadron resonance gas model, ideal thermal model, excellently reproduces the K+/π+ ratio, including its nonmonotonic horn-like structure. Therefore, we conclude that the utilization of any specific type of extensive or nonextensive statistics must be strictly aligned to the statistical nature of the underlying ensemble, otherwise, the price to be paid becomes unexpectedly huge. Such a universal conclusion apparently goes beyond the reproduction of the horn-like structure of K+/π+ ratio or particle physics to all possible statistical implications either on stock markets or economy or sociology or biological evolution or weather prediction, etc. This brings about the great deal of the present paper.