Ultra-relativistic nonthermal power-law ensembles: Cosmic-ray electrons and positron fraction

Thermodynamically stable ultra-relativistic power-law distributions are employed to model the recently measured cosmic-ray electron flux and the positron fraction. The probability density of power-law ensembles in phase space is derived, as well as an extensive entropy functional. The phase–space measure is transformed into a spectral number density, parameterized with the Lorentz factor of the charges and quantized in Fermi statistics. Relativistic power-law ensembles admit positive heat capacities and compressibilities ensuring mechanical stability as well as positive root mean squares quantifying thermodynamic fluctuations. The wideband spectral fitting of dilute nonthermal electron–positron plasmas with ultra-relativistic power-law densities is explained.