# Partition function and thermodynamic parameters of the all-particle cosmic-ray flux

## Abstract

The all-particle cosmic-ray energy spectrum is studied in the 1 GeV–10^{11} GeV interval, the relativistic nuclei being treated as a free multi-component gas in stationary non-equilibrium. A phase-space derivation of the spectral number density, partition function and entropy is given, and an analytic expression for the flux density of the all-particle spectrum is semi-empirically obtained from a wideband spectral fit. The all-particle spectrum is the additive superposition of four strongly overlapping peaks with exponential cutoffs at the spectral breaks. The analytic flux density covers the mentioned interval ranging over eleven decades and accurately reproduces the spectral fine-structure, such as two weak spectral breaks between knee and ankle emerging in the IceTop-73 and KASCADE-Grande data sets. In the low-energy range below 10^{4} GeV, the all-particle flux is approximated by adding the proton and helium flux densities obtained from fits to the AMS-02 and CREAM spectra, the contribution of heavier nuclei being negligible in this energy range. Estimates of the thermodynamic parameters (number count, internal energy, entropy and pressure) of the all-particle flux and the partial fluxes generating the spectral peaks are derived.

## Keywords

- Spectral breaks in the all-particle cosmic-ray flux;
- Wideband spectral fits with multiplicative flux densities;
- Decomposition of wideband spectra into spectral peaks;
- Partition function of relativistic stationary non-equilibrium ensembles;
- Entropy of cosmic-ray nuclei;
- Flux densities of proton and helium spectra

## 1. Introduction

We develop a statistical model of the all-particle cosmic-ray spectrum in the interval in terms of a multi-component gas of relativistic nuclei in stationary non-equilibrium. We derive an analytic expression for the spectral flux density, perform a wideband fit to the all-particle energy spectrum and estimate the thermodynamic parameters such as number count, internal energy, pressure and entropy. The flux density admits a multiplicative spectral kernel with factors defined by the spectral breaks and interconnecting slopes. The factorizing kernel allows us to systematically reconstruct the measured spectrum over an extended energy interval and is capable of resolving even minor spectral breaks. The flux density determines the spectral number density of the nuclei from which the partition function and the thermodynamic variables are derived.

The measured flux is nearly isotropic and stationary, which suggests to treat cosmic-ray nuclei as a relativistic gas of non-interacting particles, their Coulomb interaction being negligible. The spectral distributions (flux and number densities) introduced in this paper are derived in analogy to the Maxwell, Planck and Fermi equilibrium distributions. The stationary non-equilibrium densities employed in the spectral fits are a generalization of the relativistic Maxwell distribution. The deviation from equilibrium is quantified by the spectral kernel which is a multiply broken power law with smooth transitions. The parameters defining this kernel can be extracted from a spectral fit to the all-particle spectrum, without the need to specify the sources and quantify the production, acceleration, scattering and absorption mechanisms of high-energy cosmic rays.

The high-energy spectral fit of the analytic flux density to the all-particle spectrum in the range is based on data points from Tibet-III [1], IceTop-73 [2] and [3], KASCADE-Grande [4], Auger [5] and Telescope Array [6]. The exponential decay of the flux density above 10^{10} GeV is caused by the Boltzmann factor; the decay exponent, defining the temperature of the relativistic non-equilibrium gas (ultra-relativistic in this region), is obtained from the spectral fit. The all-particle spectrum is an additive superposition of four spectral peaks which can iteratively be reconstructed from the multiplicative spectral kernel.

In the mildly relativistic low-energy regime , we approximate the all-particle flux by adding the flux densities of protons and helium nuclei, as heavier nuclei constitute only about one percent of the total flux in this interval. The proton and helium fluxes in the range are inferred from spectral fits to AMS-02 [7] and [8] and CREAM [9] data sets. The multiplicative spectral representation of the flux density allows us to perform the high- and low-energy fits separately, and it provides an analytic interpolation of the all-particle flux in the decade not yet covered by data points.

The spectral flux density used in the wideband fit of the all-particle spectrum is classical, admitting a partition function which can be assembled in relativistic phase space. We also sketch the quantization of the flux and number densities, partition function and thermodynamic variables, and find the conditions for the classical limit to apply. The factorizing flux density of the all-particle spectrum admits a decomposition into four partial fluxes which produce spectral peaks intersecting at the spectral breaks, and we calculate their thermodynamic parameters.

In Section 2, we introduce the flux densities employed in the spectral fits (performed in Section 5). Starting with the spectral number density of a free relativistic gas in stationary non-equilibrium, we derive the flux density as well as integral representations of the partition function and the thermodynamic variables of number count, internal energy, entropy and pressure. The deviation of the number density from the relativistic Maxwellian equilibrium density is quantified by its factorizing spectral kernel, a multiply broken power-law with parameters to be determined from spectral fits. In this way, we arrive at a product representation of the flux density suitable for wideband fits covering multiple spectral breaks.