Tomaschitz, R.
(2017). Weibull thermodynamics: Subexponential decay in the energy spectrum of
cosmic-ray nuclei, *Physica A* **483**, 438-455, DOI:
10.1016/j.physa.2017.03.034

**Abstract (**ScienceDirect**, **SAO/NASA ADS**)**

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 1 GeV–10^{11} GeV
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*/(*k*_{B}*T*))^{σ}) and
permit a precise determination of the cutoff temperature *k*_{B}*T*=(2.5±0.1)×10^{10
}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/*T*_{eff},
*k*_{B}*T*_{eff}≈16.1GeV, 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*/(*Nk*_{B}*T*_{eff})≈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<*C _{V}*<

05.20.Gg Classical ensemble theory

05.70.Ce Thermodynamic functions and equations of state

95.30.Tg Thermodynamic processes, conduction, convection, equations of state

98.70.Sa Cosmic rays

description: Roman
Tomaschitz (2017) Weibull thermodynamics: Subexponential decay in the energy
spectrum of cosmic-ray nuclei, *Physica A ***483**, 438.

**Keywords:** Subexponential
spectral decay; Entropy of stationary non-equilibrium ensembles; Weibull
entropy and excess entropy; Multiply broken power-law densities with Weibull
cutoff; Heat capacities of a non-equilibrated gas mixture; Thermodynamics of
cosmic-ray nuclei

**Highlights**

A spectral fit to the ultra-high energy flux of cosmic-ray nuclei exhibits subexponential Weibull decay.

The spectral number density of the nuclear gas is a multiply broken power-law distribution with Weibull cutoff.

The thermodynamics of a stationary non-equilibrium gas of relativistic nuclei is developed, starting with the entropy functional in phase space.

The heat capacities, compressibilities and the expansion coefficient of a relativistic gas mixture in stationary non-equilibrium are derived.

Estimates of temperature, number count, internal energy and pressure of the all-particle cosmic-ray flux.

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