An appropriate statistical approach for nonequilibrium particle production

Abstract: The incapability of thermal models to accurately reproduce the horn-like structure of the Kaon-to-pion ratio measured at AGS, SPS, and low RHIC energies, as well as confirmed in the beam energy scan program, has long been a persistent problem. This issue is believed to have arisen due to the inappropriate application of statistics, particularly the extensive additive Boltzmann-Gibbs (BG) statistics. The assumption that the analysis of particle production, a dynamic nonequilibrium process, should be primarily conducted using extensive BG or nonextensive Tsallis statistics, has proven to be an unsuccessful approach that has been followed for several decades. By employing generic (non)extensive statistics, two equivalence classes $(c,d)$ emerge, thereby undermining the validity of any ad hoc assumption. Consequently, the degree of (non)extensivity exhibited by the statistical ensemble is determined by its own characteristics. This encompasses both extensive BG statistics, characterized by $(1,1)$, and nonextensive Tsallis statistics, characterized by $(0,q)$. The energy dependence of light-, $\gamma_q$, and strange-quark occupation factor, $\gamma_s$, suggests that the produced particles are most appropriately described as a nonequilibrium ensemble. This is evidenced by a remarkable nonmonotonic behavior observed in the $\mathrm{K}^+/\pi^+$ horn, for instance. On the other hand, the resulting equivalence classes $(c,d)$ are associated with a generic nonextensivity related to extended exponential and Lambert-$W_0$ exponentially generating distribution function, which evidently arise from free, short- and long-range correlations. The incorporation of generic nonextensive statistics into the hadron resonance gas model yields an impressive ability to rightfully reproduce the nonmonotonic $\mathrm{K}^+/\pi^+$ ratio.