Description of the Energy Spectrum of Cosmic Rays Using Superstatistics and Gamma Distribution

Although the origin of cosmic rays and spectral breaks remains a controversial issues, it is evident that the energy spectrum of cosmic rays does not follow a Gaussian distribution and therefore, cannot be described using ordinary statistical mechanics. In this study, we apply Beck-Cohen superstatistics, a generalized framework of statistical mechanics. In this model, the main idea is that the energy spectrum of cosmic rays results from many high-energy scattering processes, each with a different local temperature \(\beta\) $\beta$ in the local scattering volume. By considering the temperature fluctuations, one can not only obtain Tsallis statistics but also an infinite set of statistics. In this paper, we utilize a gamma distribution to describe these temperature fluctuations, employing shape and scale parameters as essential tools. Our approach facilitates the derivation of a novel relationship for the entropy index q as a function of energy, which has not been previously explored in the context of cosmic rays. The effective temperature is estimated to be 100 MeV, aligning with the generalized Hagedorn temperature. Importantly, our theoretical results show strong agreement with available experimental data.