Weibull thermodynamics: Subexponential decay in the energy spectrum of cosmic-ray nuclei

The spectral number density of cosmic-ray nuclei is shown to be a multiply broken power law with subexponential spectral cutoff. To this end, a spectral fit is performed to data sets covering the 1GeV−1011GeV interval of the all-particle cosmic-ray spectrum. The flux points of the ultra-high energy spectral tail measured with the Telescope Array indicate a Weibull cutoff exp(−(E∕(kBT))σ) and permit a precise determination of the cutoff temperature kBT=(2.5±0.1)×1010 GeV and the spectral index σ=0.66±0.02. Based on the spectral number density inferred from the least-squares fit, the thermodynamics of this stationary non-equilibrium system, a multi-component mixture of relativistic nuclei, is developed. The derivative of entropy with respect to internal energy defines the effective temperature of the nuclei, S,U=1∕Teff,kBTeff≈16.1 GeV, and the functional dependence between the cutoff temperature in the Weibull exponential and the effective gas temperature is determined. The equipartition ratio is found to be U∕(NkBTeff)≈0.30. The isochoric and isobaric heat capacities of the nuclear gas are calculated, as well as the isothermal and adiabatic compressibilities and the isobaric expansion coefficient, and it is shown that this non-equilibrated relativistic gas mixture satisfies the thermodynamic inequalities 0