# The stationary non-equilibrium plasma of cosmic-ray electrons and positrons

## Highlights

- •
The thermodynamic variables of the cosmic-ray electron–positron plasma are calculated.

- •
Spectral fits to the AMS-02 and HESS GeV–TeV electron and positron fluxes are performed.

- •
A semi-empirical phase-space reconstruction of the partial probability densities is carried out.

- •
Partition function & entropy of a relativistic plasma in stationary non-equilibrium are derived.

- •
The positron fraction extrapolated to TeV energies shows a broad peak & exponential decay.

## Abstract

The statistical properties of the two-component plasma of cosmic-ray electrons and positrons measured by the AMS-02 experiment on the International Space Station and the HESS array of imaging atmospheric Cherenkov telescopes are analyzed. Stationary non-equilibrium distributions defining the relativistic electron–positron plasma are derived semi-empirically by performing spectral fits to the flux data and reconstructing the spectral number densities of the electronic and positronic components in phase space. These distributions are relativistic power-law densities with exponential cutoff, admitting an extensive entropy variable and converging to the Maxwell–Boltzmann or Fermi–Dirac distributions in the non-relativistic limit. Cosmic-ray electrons and positrons constitute a classical (low-density high-temperature) plasma due to the low fugacity in the quantized partition function. The positron fraction is assembled from the flux densities inferred from least-squares fits to the electron and positron spectra and is subjected to test by comparing with the AMS-02 flux ratio measured in the GeV interval. The calculated positron fraction extends to TeV energies, predicting a broad spectral peak at about 1 TeV followed by exponential decay.

## Keywords

- Stationary non-equilibrium distributions;
- Cosmic-ray electron–positron plasma;
- Relativistic statistical ensembles;
- Power-law densities with exponential cutoff;
- Nonthermal ensemble averaging;
- Classical & quantum partitions with extensive entropy

## 1. Introduction

We study the statistical mechanics of the cosmic-ray electron–positron plasma, based on high-precision spectra obtained with the Alpha Magnetic Spectrometer (AMS-02) [1] and [2]. We demonstrate that this plasma can be treated, in close analogy to the photon gas of the cosmic microwave background radiation, as a primordial electron gas in stationary non-equilibrium. A primordial origin is also suggested by the high isotropy of the observed fluxes. Primordiality of the cosmic-ray electron–positron plasma is tantamount to the material realization of a universal cosmic reference frame as the rest frame of a relativistic gas of massive particles. An alternative approach is to use relativistic kinetic theory, but to arrive at quantitative densities suitable for spectral fitting, one has to specify production mechanisms and interaction processes with cosmic matter and radiation fields which are uncertain [3].

We perform spectral fits to the AMS-02 electron and positron fluxes and reconstruct, from the inferred spectral densities, the distribution functions of the electronic and positronic plasma components. The partial number densities are relativistic non-equilibrium distributions, exponentially cut power-law densities [4] and [5] which converge to the Maxwell–Boltzmann distribution for low particle velocities, the Coulomb interaction being negligible. We calculate the classical partition function, by ensemble averaging in phase space, and the entropy function which is an extensive variable. We then quantize the grand partition function and show, based on the fugacity obtained from the spectral fits, that the classical limit is realized. Finally we calculate the positron fraction and give estimates of the thermodynamic parameters of this low-density high-temperature plasma.

In Section 2, we consider a relativistic plasma in stationary non-equilibrium and relate the classical spectral number density to the empirical flux density obtained from spectral fits to the measured electron and positron fluxes (performed in Section 5). In Section 3, we derive the thermodynamic variables of the electronic and positronic plasma components in phase space, based on probability distributions inferred from the spectral fits, in particular the electronic/positronic energy and entropy densities.

In Section 4, we explain the quantization of the classical nonthermal partition function and demonstrate that the classical limit applies to the cosmic-ray electron–positron plasma due to the small fugacity. The formalism developed in Sections 2, 3 and 4 is based on relativistic dispersion relations and designed to be practically suitable for spectral fits in any energy range. Foundational aspects of relativistic statistical mechanics and thermodynamics such as the arrow of time and entropy are discussed in Refs. [6], [7], [8] and [9]. Recent applications of relativistic statistical mechanics include the quark–gluon plasma [10] and [11], plasmas in gravitational fields [12], gases of neutral particles with long-range spin–spin interaction [13] and condensation effects in relativistic Bose–Einstein gases [14]. Kinetic theory with relativistic dispersion relations, in particular the relativistic Boltzmann equation in Grad’s moment expansion and Chapman–Enskog approximation, is discussed in Refs. [15] and [16] and a relativistic version of the Fokker–Planck equation in Ref. [17].

In Section 5, we perform least-squares fits to the AMS-02 and HESS [18] electron and positron spectra. The measured spectra are located in the GeV and low TeV range, where we can use the ultra-relativistic approximation of the spectral densities derived in Sections 2, 3 and 4, which means to drop the mass term in the electronic dispersion relation. The ultra-relativistic electronic/positronic flux densities obtained in this way are then used to assemble the positron fraction, which has been measured by the AMS-02 Collaboration in the GeV range [19], thus providing a test of the spectral number densities on which the thermodynamic variables are based. Extrapolation of the analytic formula for the positron fraction into the TeV range suggests an extended spectral peak centered around 1 TeV and exponential decay above 10 TeV.

As mentioned, this primordial approach to the cosmic-ray electron–positron plasma is alternative to the use of kinetic equations which require the assumption of specific electron/positron production mechanisms [20]. A possible mechanism is electron/positron emission caused by interaction of high-energy protons or heavier cosmic-ray nuclei with the interstellar medium [3] and [21]. Quantitative models thereof turn out to be inconsistent with the pronounced rise of the positron fraction, so that additional positron sources have to be invoked, for instance, discrete high-energy sources such as pulsar magnetospheres or the shocked plasmas of supernova remnants [22], even if isotropy is somewhat compromised. Other authors prefer positron production via decay or annihilation of a variety of hypothetical dark matter particles at TeV energies [23] which could exist in galactic halos.

In Section 5, we also give estimates of the specific energy and number densities, entropy production, temperature, fugacity and chemical potential of the electronic/positronic plasma components. We calculate the specific densities of the actually measured electrons/positrons in the energy range above 1 GeV up to low TeV energies where an exponential cutoff occurs. We then restore the electron mass in the dispersion relation and extrapolate the analytic spectral densities obtained from the spectral fits to lower energies. In this way, we can predict the specific energy, number and entropy densities of the complete statistical ensemble comprising all gas particles by integrating the electronic/positronic spectral densities down to the electron mass. In Section 6, we present our conclusions.

## 2. Relativistic plasma in stationary non-equilibrium

We start with the classical number density of a free electron plasma,

The number density (2.1) is assembled with the integration measure and the dispersion relation for free relativistic particles , so that . The normalization factor s/(2π)^{3} arises from box quantization in the continuum limit, see (4.4). The quantized spectral density of which (2.1) is the classical limit is discussed in Section 4. Density (2.1) is thus assembled as , with the normalized spectral function G(E)=g(E)/g(m). The non-relativistic limit of is thermal since , where υ≪1
is the particle speed parametrizing the non-relativistic
Maxwell–Boltzmann distribution, whose chemical potential is obtained
from the relativistic potential by subtraction of the rest mass m.

The spectral flux density is found by multiplying the number density with the group velocity, . The flux density per steradian is thus

where we have indicated the units used in the spectral fits. We rescale with E^{3}to find The spectral fits in Section 5 are performed with the flux density [24] and [25] where the exponents α

_{1},β

_{1},γ

_{1},β and amplitudes A

_{Φ},b

_{1},c

_{1}are fitting parameters. The factor 1−m

^{2}/E

^{2}stems from the electronic dispersion relation which enters both in the group velocity and the integration measure stated above. (We have put ħ=1, so that the momentum variable p of free particles coincides with their wavenumber k, cf. Section 4.) In the ultra-relativistic limit, one assumes that the mass/energy ratio is small compared to 1 so that it can be dropped, p∼E. The spectral fits in Section 5 are performed in the GeV interval, the ultra-relativistic regime, where the squared ratio is smaller than 10

^{−6}, so that the flux density becomes effectively independent of the electron mass at GeV energies and above. In the relativistic formalism developed in Sections 2, 3 and 4, we maintain this ratio in the dispersion relation, so that the spectral densities defining the thermodynamic variables remain valid for particle energies close to the rest mass and in the intermediate regime.

In flux density (2.4), we specified the spectral function g(E) in (2.3) by the ansatz

and the flux amplitude A_{Φ}reads Here, we have restored the units, put s=2 (spin multiplicity), and used GeV units for the electron mass and the coefficients c

_{1}and b

_{1}in (2.5).

The number of particles in a volume V with energies exceeding is calculated as

and the total energy of particles with energies above is In Section 5, we will use the lower edge of the measured spectrum as cutoff energy, putting , so that specific densities like U/V refer to the observed GeV and low-TeV electrons/positrons. We will also extrapolate the analytic spectral measure obtained from the spectral fits to lower energies and identify in (2.8) with the electron mass m to obtain the total specific densities covering all particle energies above the rest mass. In this case, the electronic specific energy density increases to as compared with a density of comprising electrons exceeding the experimental threshold energy of 1 GeV, see Table 2 in Section 5. (For comparison, the specific energy density of the primordial photon background is .) In Table 2, we record the specific particle, energy and entropy densities of electrons/positrons in the measured GeV–TeV interval () as well as the total specific densities based on the extrapolated spectra (), also see the discussion at the end of Section 5.Entropy is an extensive quantity proportional to V,

where we can replace the logarithm of the classical partition function by the particle number N.To demonstrate thermodynamic stability, we first show that the inequalities 0≤C_{V}≤C_{P} hold for the isochoric and isobaric heat capacities. To this end, we note, cf. (2.7) and (2.8),

_{,β})

^{2}≤KK

_{,β,β}implies (logK)

_{,β,β}≥0 (subscript commas indicate differentiation with respect to β), so that the above inequalities hold.

We also have to show mechanical stability, 0≤κ_{S}≤κ_{T}, where κ_{S} and κ_{T} denote adiabatic and isothermal compressibility. Since , cf. (2.7), the thermal equation of state reads P=N/(βV), and the isothermal compressibility is thus κ_{T}=−(1/V)∂V(β,P,N)/∂P=1/P. The adiabatic compressibility is related to the heat capacities by κ_{S}=κ_{T}C_{V}/C_{P}, which implies the stated inequalities since κ_{T}>0 and C_{P}>C_{V}. This holds true for the electronic and positronic plasma components alike.

## 3. Thermodynamic variables in phase space

We consider the phase space of n electrons with measure

k=1,…,n. The integrations range over a volume V, and the integrations refer to the interval , cf. (2.7) and after (2.1). The normalization (2π)^{3n}is suggested by the quantized spectral number density (4.7) in the classical limit. The n-particle probability distribution reads The factorial accounts for indistinguishability, s is the spin degeneracy and ξ a normalization constant. The number densities listed in Table 2 are very low and the temperature of the electronic/positronic components is sufficiently high to neglect the Coulomb interaction; also see Section 6 for a numerical estimate. Otherwise we would have to incorporate the Coulomb potential in the probability density and use, for instance, the Debye–Hückel approximation to calculate the thermodynamic variables [26].

The phase-space probability is normalized to one,

where summation and integration can be interchanged and the integrals factorize. This normalization is achieved by identifying ξ in (3.2) with the logarithm of the partition function , cf. (2.7). Thus, The expectation values of particle number and internal energy read The spectral number density (2.1) is identified by way of (3.5). The entropy stated in (2.9) is assembled in phase space as which can be written as expectation value,## 4. Quantized partition function of the electronic and positronic plasma components

The quantized version of the classical partition function in (2.7) is obtained by a trace calculation in fermionic occupation number representation [25] and [27],

The particle number operator is , where the are fermionic creation/annihilation operators. The energy operator reads and the empirical spectral function f(E) in (2.1) defining the deviation from equilibrium is quantized as . The Hermitian number operators are diagonal, , and the fermionic occupation numbers attached to a wave vector can take the values zero and one. Electron energy and wavenumber are related by the dispersion relation , cf. after (2.4). The wavenumber of the lower energy threshold in (2.7) and (2.8) is . We will employ box quantization, discretizing the wave vectors as , so that summations are taken over integer lattice points , corresponding to periodic boundary conditions on a box of size V=L^{3}.

With these prerequisites, trace (4.1) can be evaluated as

By taking the logarithm, we find Performing the continuum limit L→∞ and taking into account the spin degeneracy, we can replace the summation in (4.3) by the integration , The integration measure is parametrized like indicated after (2.1), The quantized spectral number density is identified by means of the particle number, The nonthermal Fermi density (4.7) can be replaced by the Boltzmann density in (2.1) if . As the inverse fugacity turns out to be very large, cf. Table 2, we can use the classical limit (2.1) to calculate the thermodynamic parameters of the electronic and positronic components, cf. (5.3).## 5. Spectral fits to high-energy cosmic-ray electron and positron fluxes

The spectral fits in Fig. 1 and Fig. 2 are performed in the GeV and TeV range, the ultra-relativistic regime, so that we can drop the factor (1−m^{2}/E^{2}) in the flux density, see the discussion following Eq. (2.4),

_{1},β

_{1},γ

_{1}and amplitudes A

_{Φ},b

_{1},c

_{1}as well as the temperature parameter β determining the exponential cutoff; they are listed separately for the electronic and positronic plasma components in Table 1. Even though the spectral maps in Fig. 1 and Fig. 2 look quite different at first sight, they can both be reproduced with flux density (5.1). In Fig. 3, we compare the analytic flux ratio obtained from the spectral fits to the measured positron fraction. We also depict the asymptotic limit of this ratio, (E in GeV units), which exhibits exponential decay since the positron flux decays faster than the electron flux, cf. Table 1. The spectral fits in Fig. 1 and Fig. 2 are based on the AMS-02 and HESS 2008 flux points [1] and [18]. There have been several other experiments attempting to measure the cosmic-ray electron and positron fluxes above 1 GeV and the positron fraction [28], [29], [30], [31] and [32]. They are not included in the least-squares fits, as they define different spectral curves which are only marginally compatible with the AMS-02 data sets which admit smaller error bars and extend to higher energies [1].

- Full-size table

- Full-size table

The classical spectral number density reads, cf. (2.1) and Refs. [33] and [34],

where g(E;α_{1},β

_{1},γ

_{1},b

_{1},c

_{1}) denotes the spectral function (2.5); the indicated parameters are obtained from the χ

^{2}fits of the electron and positron fluxes and are listed in Table 1, as are temperature and fugacity , the latter is extracted from the flux amplitude A

_{Φ}, cf. (2.6). We denote specific thermodynamic densities by lower case letters with a hat; the specific number and energy densities read, cf. (2.7) and (2.8), where is the relativistic spectral density (5.2). The specific entropy density rescaled with the Boltzmann constant is assembled as, cf. (2.9), The specific particle, energy and entropy densities of the electronic and positronic plasma components are recorded in Table 2, together with temperature, fugacity and chemical potential. We list the specific densities of the observed GeV–TeV electrons and positrons and also give estimates of the total specific densities covering the complete spectrum by extrapolating the spectral measure (5.2) to lower energies and identifying the cutoff energy in (5.3) with the electron mass, . That is, the specific densities listed in the first and third row of Table 2 comprise electrons/positrons measured by the AMS-02 experiment in the energy interval of 1 GeV up to a few TeV where the spectral densities are exponentially cut; the lower integration boundary in (5.3) is thus . The specific densities listed in the second and fourth row of Table 2 are also based on the analytic relativistic spectral densities (5.2) derived from the same spectral fits, but in this case we use the electron mass 0.51 MeV as lower integration boundary in (5.3), so that the densities take account of all gas particles including those in the MeV interval which cannot be resolved with the AMS-02 spectrometer. The specific densities of internal energy, number count and entropy of the positronic plasma component are by one order lower than the electronic counterparts, whereas the estimates of temperature and chemical potential are quite comparable. This holds true for the observed densities (row 1 and 3 of Table 2) as well as for the predicted densities of the total electronic/positronic plasma components recorded in row 2 and 4.

## 6. Conclusion

We developed a statistical description of the cosmic-ray electron–positron plasma based on stationary non-equilibrium distributions. The latter are inferred from spectral fits to the electron and positron spectra measured by the AMS-02 experiment in the GeV interval and the Cherenkov telescope HESS in the low TeV range. In particular, we do not need to make hypothetical assumptions on sources and production mechanisms of high-energy cosmic-ray electrons/positrons. Their spectral number densities are semi-empirically reconstructed from the measured electron and positron fluxes as classical ensemble averages in phase space.

The
cosmic-ray electron–positron plasma is relativistic and nonthermal, the
Coulomb interaction can be neglected due to the low particle density
and high temperature. The criterion for this approximation is a small
ratio , where is the interparticle distance and e^{2}/(4π)≈1/137 the fine-structure constant [35] and [36]. In the case of cosmic-ray electrons and positrons, this ratio is of order 10^{−23}. Temperature and the specific number density of the non-equilibrated plasma components are listed in Table 2. Quantum corrections to the classical partition function are also negligible, cf. after (4.7).

The entropy of the cosmic-ray electron–positron plasma is an extensive quantity, even though the underlying electronic/positronic probability distributions are nonthermal. We also checked the positivity of the heat capacities and compressibility, which ensures the thermodynamic stability of each plasma component implying positive root mean squares of the thermodynamic variables. The thermal Maxwell–Boltzmann distribution is recovered in the non-relativistic limit of the spectral number density (2.1).

To summarize the approximations, cosmic-ray electrons and positrons at GeV energies and beyond constitute a relativistic gas, so that one has to use relativistic dispersion relations. As the particles are charged, they constitute a two-component plasma, but since the gas is dilute and thus the interparticle distances large according to the number densities in Table 2, one can ignore Coulomb interaction and Debye screening, see the estimate above. As the fugacity is low, the quantum distribution (4.7) can be approximated by its classical limit. In the spectral fits in Section 5, the ultra-relativistic approximation is used, because the electronic mass/energy ratio is negligible in the GeV band, cf. after (2.4). In the calculation of the specific thermodynamic densities of the electronic and positronic gas components, cf. Table 2, we restored this ratio in the electronic dispersion relation, since the fitted spectral densities determining the specific densities are extrapolated to lower energies, see the remarks after (5.4).

We calculated the positron fraction using the parameters obtained from the spectral fits in Fig. 1 and Fig. 2, and compared it with the AMS-02 fraction measured in the GeV range. As the analytic representation (5.1) of the electronic/positronic flux densities also applies at TeV energies and since the temperature of the positronic plasma component is lower than of the electrons, cf. Table 1, we predict a broad spectral peak of the positron fraction at around 1 TeV which is followed by exponential decay, see after (5.1) and Fig. 3.

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