Roman Tomaschitz 2013 *EPL* **104** 19001

doi:10.1209/0295-5075/104/19001

Copyright © EPLA, 2013

Received 31 July 2013,
accepted for publication 3 October 2013

Published 4 November 2013

- Share on emailEmail
- Share on facebookFacebook
- Share on twitterTwitter
- Google+1
- Share on citeulikeCiteULike
- Bibsonomy
- PDF (666 KB)
- More Sharing ServicesShare

**View usage and citation metrics for this article**
**Get permission to re-use this article**

A statistical description of the all-particle cosmic-ray spectrum is given in the *E*^{3}-scaled
flux representation in which the spectral fit is performed. The
all-particle spectrum is covered by recent data sets from several air
shower arrays, and can be modeled as three-component plasma in the
indicated energy range extending over six decades. The temperature,
specific number density, internal energy and entropy of each plasma
component are extracted from the partial fluxes in the broadband fit.
The grand partition function and the extensive entropy functional of a
non-equilibrated gas mixture with power-law components are derived in
phase space by ensemble averaging.

The high–and ultra-high–energy cosmic-ray flux (*E*^{3}-scaled flux representation employed in the wideband fit.

We sketch the basic thermodynamic formalism of classical ultra-relativistic power-law ensembles and nonthermal mixtures thereof, starting with the spectral number density parametrized with the Lorentz factor. We then define the specific number count and the internal energy of the gas components, as well as their entropy density. After this overview, we discuss the phase-space measure and probability density of nonthermal power-law ensembles, and derive the grand canonical partition function and the extensive entropy functional of multi-component power-law mixtures involving different particle species at different temperature.

We discuss the practical aspects of broadband
spectral fitting with flux densities of classical unequilibrated
(stationary non-equilibrium) power-law mixtures in the
ultra-relativistic regime, and perform the spectral fit of the
all-particle cosmic-ray spectrum above

An ultra-relativistic classical gas mixture composed of nonthermal power-law components is defined by the spectral number density

parametrized by the Lorentz factor *γ*. The positive spectral functions *M*-component mixture is composed of particle species *m*_{i} is the mass and *Z*_{i} the (positive or negative) integer charge number (in multiples of the electron charge *e* > 0) of the particles in the respective component *m*_{i} is a shortcut for *m*_{i} *c*^{2}, so that *s*_{i} is the spin multiplicity and

where *a*_{i} is positive. Spectral functions of this type are thermodynamically stable [18] and suitable to model wideband spectra [20–22]. Quantized power-law ensembles with

The total particle number *V* are determined by the partial densities

The lower integration boundary is the cutoff Lorentz factor

The partition function of a gas component

Apparently

obtained from an ensemble average of power-law partitions, see the next section.

↑ CloseWe give a probabilistic derivation of the classical spectral number density *n* particles of a gas component *M*-component mixture,

The *V*, and the

The probability density on the *n*-particle states of a gas component

The factor *s*_{i} for the spin multiplicity of this component.

The normalization condition for the phase-space probability of an *M*-component mixture is

where summation and integration sign can be interchanged, as well as the integration and product signs. The

for each gas component

The expectation values of particle number and internal energy of a gas component

where

which is the expectation value

The partial entropies *S*_{i} stated in (5) are recovered from (13) by substitution of the averages (10) and (11).

We write the components of the total spectral number density

with spectral function *m*_{i} stands for *m*_{i} *c*^{2}, *s*_{i} is the spin degeneracy of component

where we introduced the shortcuts

We also note *E* and *m*_{i} in eV units, and the units chosen for length, time and temperature are, respectively, meter, second and kelvin.

The spectral flux density of a gas component is found by multiplying the number density (15) with the particle velocity

Density *E*. Wideband spectra are assembled by adding the partial flux densities of the gas components, *E*^{k}. The rescaled density

where we defined the amplitude

using dimensionless quantities, cf. after (16). In the ultra-relativistic regime, we can drop the factor

The chemical potential

and

where the lower energy cutoff (integration boundary) is related to the Lorentz factor by *E* to the integrand in (21). The specific densities *m*_{i} in the ultra-relativistic regime

where we have put

**Table 1:.**
Thermodynamic parameters of the ultra-relativistic number densities *T*_{i}, fugacity *z*_{i} and chemical potential

i |
||||||

1 | 745 | −113 | ||||

2 | 18 | −131 | ||||

3 | 0.031 | −136.5 | ||||

mix | 763 | – | – | – |

The fit of the all-particle cosmic-ray spectrum in fig. 1 is performed with the ultra-relativistic limit of the flux density *E*^{3}, defining

The fitting parameters in (23) are the power-law exponents