Nonextensive Statistical Mechanics

7.1 Physics 225hypergeometric functions) with the experimental data is quite impressive for theentire range of transverse hadronic momenta. The phenomenological value of qslightly increases from q = 1toq ≃ 1.2 (the asymptotic value 11/9 has beensuggested [406] for the very large energies) when the center-of-mass collisionalenergy increases from 14 to 161 Gev. See [405] for a possible physical origin ofthis effect.Many other high-energy multiparticle production processes (from collisions suchas pp, p ¯p, Au+ Au, Cu+Cu, Pb+ Pb, etc.) have been analyzed along related lines[407–414]. The values of q that emerge (from the BRAHMS, STAR, PHENIX data,for instance) are systematically close to the case discussed above, typically in therange 1 < q < 1.2 . The nonzero values of q − 1 are frequently interpreted in termsof sizeable temperature fluctuations that exist during the hadronization process (see[327, 328, 384]).7.1.2.2 Flux of Primary Cosmic RaysCosmic rays arrive to Earth within a vast range of energies, up to values close to10 20 ev. Their associated fluxes vary within impressive 33 orders of magnitude: seeFig. 7.4. This curve includes the so-called “knee” and “ankle,” at intermediate andvery high energies, respectively. It turns out that it is possible, without entering intoany specific mechanism, to provide [415] an excellent phenomenological descriptionof these data by assuming a crossover between two q-exponential distributionfunctions. The two corresponding values of q are quite close among them, and alsoclose to 11/9 ( [416]).7.1.2.3 Quantum Scattering of ParticlesEntropic bounds for scattering of spinless particles (e.g., pions) by a nucleus havebeen established and tested [417–420] with available experimental results for phaseshifts. Typical results involving 4 He, 12 C, 16 O, and 40 Ca nuclei are exhibited inFig. 7.5. Along this line, a conjugation relation naturally emerges for two relevantentropic indices, noted q and ¯q (see details in [419, 420]). This relation is given bywhich can equivalently be written as1+ = 2 , (7.2)q1¯q¯q = μνμ(q) , (7.3)where the multiplicative and additive dualities μ and ν are those defined inEqs. (4.39) and (4.40), respectively. A deeper understanding of this intriguing connectionwould be welcome.