Generalized statistical mechanics of cosmic rays

We consider a generalized statistical mechanics model for the creation process of cosmic rays which takes into account local temperature fluctuations. This model yields Tsallis statistics for the cosmic ray spectrum. It predicts an entropic index q given by q = 11/9 at largest energies (equivalent to a spectral index of α = 5/2), and an effective temperature given by 59TH , where kTH ≈ 180 MeV is the Hagedorn temperature measured in collider experiments. Our theoretically obtained formula is in very good agreement with the experimentally measured energy spectrum of primary cosmic rays. More general versions of statistical mechanics, as introduced by Tsallis [1] and further developed by many others [2, 3], have recently been successfully applied to a variety of complex physical systems. The idea is to maximize more general entropy measures than the Shannon entropy, which depend on a parameter q and which lead to generalized versions of statistical mechanics. Ordinary statistical mechanics is contained as a special case for q = 1. Interesting recent physical applications of the so-called nonextensive formalism obtained for q 6= 1 include the statistics of fully developed hydrodynamic turbulence [4], defect turbulence [5], scattering processes in ee annihilation [6, 7], heavy ion collisions [8], hadron collisions [9], and models of vacuum fluctuations at the Planck scale [10]. There is growing evidence that the nonextensive formalism, though possessing the mathematical structure of an equilibrium formalism, is physically often relevant for nonequilibrium systems with a stationary state that possess strong fluctuations of an intensive parameter [11, 12, 13]. Very recently Tsallis and Borges [14] analysed the experimentally measured energy spectrum of cosmic rays from a nonextensive point of view. They showed that a good fit of the measured spectrum can be obtained by assuming the interplay of two different nonextensive canonical distributions with two different entropic indices q1 and q2 and two different inverse temperatures β1 and β2. Moreover, Kaniadakis [15] showed that yet another type of generalized statistical mechanics (based on entropies that are neither Shannon nor Tsallis) also yields a good fit of the measured spectrum, using three fitting parameters. Both these papers represent new interesting ideas, but at the same time they use various fitting parameters which are not predicted from first principles. The question arises whether one can construct a generalized statistical mechanics model of cosmic rays that predicts the relevant parameters from first principles and that at the same time well reproduces the measured cosmic ray energy spectrum using these parameters. In this letter we consider such a generalized statistical mechanics model. The model is quite generally of relevance for particles that are created by scattering processes at very large energies (as cosmic ray particles are). The basic idea is that at very high energies the effective thermodynamic interaction volume is small and hence one expects that there are strong local temperature fluctuations. A relevant thermodynamic theory of high-energy scattering processes is the Hagedorn theory [16], which we will generalize here in the sense that we take into account local temperature fluctuations. Under reasonable assumptions on the form of the temperature fluctuations