Volume 394, 15 January 2014, Pages 110–123
Ultra-relativistic nonthermal power-law ensembles: Cosmic-ray electrons and positron fraction
- Department of Physics, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima 739-8526, Japan
- Received 13 June 2013
- Revised 11 September 2013
- Available online 9 October 2013
Relativistic power-law densities admit a stable and extensive entropy functional.
Positive heat capacities and compressibility ensure their thermodynamic stability.
Quantization of nonthermal relativistic power-law ensembles in Fermi statistics.
Spectral fitting of nonthermal plasmas with ultra-relativistic power-law densities.
Entropy, temperature and chemical potential of the cosmic-ray electron/positron flux.
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.
- Nonthermal power-law distributions;
- Stability and extensivity of entropy;
- Quantization of stationary non-equilibrium ensembles;
- Hybrid quantum ensembles;
- Ultra-relativistic electron–positron plasma;
- Spectral fitting of a two-component plasma with power-law densities;
- Cosmic-ray electron/positron flux
We study nonthermal power-law ensembles capable of reproducing the flux of ultra-relativistic particles , , ,  and  such as cosmic-ray electrons and positrons. In particular, we derive power-law distributions from empirical spectral fits which are thermodynamically and mechanically stable, admitting positive heat capacities and compressibilities. The entropy of power-law ensembles is shown to be an extensive variable, and the positivity of the root mean squares of thermodynamic variables is explicitly demonstrated, ensuring a negative second entropy differential d2S≤0 subject to the extremal condition dS=0.
The spectral number density of ultra-relativistic power-law ensembles admits a convenient parameterization by the Lorentz factor of the particles. We derive the classical number density from the phase–space probability measure, which is taken as the starting point for the quantization of classical power-law ensembles in Fermi statistics. We also briefly outline the bosonic quantization of power-law densities as well as the fermionic counterpart to electromagnetic gray-body radiation, with Lorentz factors defined by group velocity in a dispersive spacetime. Hybrid quantum distributions interpolating between Fermi and Bose statistics are derived.
In Section 2, we give an overview of the power-law densities studied, including their partition function, the basic thermodynamic variables, their variances, the equations of state, and show thermodynamic stability. In Section 3, we study classical power-law densities in phase space, their probability density and entropy functional, fluctuations of internal energy, particle number and entropy, as well as the quantization of nonthermal power-law ensembles.
In Section 4, we discuss wideband spectral fitting with ultra-relativistic power-law densities. The power-law densities studied here are aimed at this purpose, being thermodynamically stable empirical distributions sufficiently general to reproduce the multiple spectral peaks of broadband spectra. We show how to extract temperature, fugacity and chemical potential, as well as the power-law amplitudes and exponents from the measured particle flux. We also discuss energy scales and conditions for the applicability of the classical limit of the distributions.
In Section 5, we perform a spectral fit to the cosmic-ray electron flux, employing an ultra-relativistic classical power-law density. We use GeV data sets collected with the PAMELA and Fermi satellites ,  and , extended into the low TeV region by an atmospheric Cherenkov array ,  and . Based on the electronic number density (obtained from the spectral fit) and the recently measured positron fraction, we calculate the spectral number density of the cosmic-ray positron flux in the 1 GeV–3 TeV interval. Finally we estimate the specific energy and number densities, as well as temperature and entropy production of ultra-relativistic cosmic-ray electrons/positrons, which constitute a nonthermal two-component plasma, dilute and at high temperature. In Section 6, we present our conclusions.
2. Fermionic power-law distributions
2.1. General outline: spectral number density, partition function, entropy
Fermionic power-law ensembles are defined by the distributions (number densities)
The particle number in a volume V isEcut=mγcut, where γcut≥1 is the cutoff Lorentz factor. The partition function ZF is stated in (2.6). The ultra-relativistic limit of dρF(γ) is obtained by replacing in (2.1), which applies if γcut≫1. The thermal Fermi–Dirac distribution is recovered with δ=0,g(γ)=1 and γcut=1 as the lower integration boundary of the partition function. Here, we study nonthermal averages with real power-law index δ and empirical spectral function g(γ) extracted from spectral fits of measured flux densities.
The spectral number density (2.1) is based on the dispersion relation . The integration in momentum space is parameterized as , so that the particle density (2.1) is compiled as
Remark. We may consider a more general dispersion relation, , with energy-dependent dimensionless permeabilities and resulting in a dispersive power-law density, a massive counterpart to photonic gray-body radiation. The integration in (2.3) is then over intervals in which the radicand of p(γ) is positive, and the normalization is done with the smallest possible Lorentz factor, instead of g(1). We still have γ=E/m, and the Lorentz factor γ can be parameterized by group velocity via m/υ=dp/dγ. Here, we use vacuum permeabilities, .
The internal energy (total energy of particles with Lorentz factor exceeding the lower threshold γcut) is. The grand partition function reads, cf. (2.3) and (2.5), S/V,U/V and N/V, the volume factor drops out in (2.7), since logZF∝V, so that S is homogeneous. We can also replace the partition function in (2.7) by the pressure variable P,logZF=PVβ/m.
The classical limit of the partition function (2.6) is obtained by expanding the logarithm,dρB is the Boltzmann power-law density dρF in (2.1) and (2.2), apparently valid for . High- and low-temperature expansions of classical power-law ensembles based on dρB in (2.9) with g(γ)=1 and s=1 can be found in Ref. .
The spectral density of the non-relativistic Fermi gas,(2.1) as γ=(1−υ2)−1/2≈1+υ2/2. The power-law factor G(γ) does not enter in leading order in the υ2 expansion, G(γ)∼1, and the rest mass can be absorbed in the fugacity, z0=e−α0,α0=α+β. The non-relativistic chemical potential μ0 is obtained by subtracting the rest mass from the relativistic potential, μ0=−α0m/β=μ−m. We find the particle count υ is dimensionless, measured in units of c. The classical Maxwell distribution is obtained by expanding the denominator of (2.10) and the logarithm in (2.12). The non-relativistic limit of the power-law densities (2.1) is always a thermal distribution, since G(γ)∼1 in leading order.
2.2. Thermodynamic stability: positivity of specific heat and compressibility
For the remainder of this section, we consider the classical limit based on the partition function (2.8) and density dρB in (2.9). In this limit, we can identify N=logZB, cf. (2.3), so that P=mN/(βV) is the thermal equation of state, cf. after (2.7). We may then explicitly solve (2.8) for α(β,N/V) and substitute this into the classical limit of the internal energy (2.5) and entropy (2.7) to obtain the U(β,N,V) and S(β,N,V) representations thereof:g(γ) is the power-law ratio in (2.2). We write N(β,α,V) in (2.3) and U(β,α,V) in (2.5) as
The entropy (2.7) reads in the S(β,N,V) representationK(β) in (2.14). The free energy F(β,N,V) is found via (2.16) and (2.17), F=U−mS/(kBβ). The isothermal compressibility follows from the thermal equation of state, κS=κTCV/CP. Thus, if 0≤CV≤CP is satisfied, this also implies mechanical stability 0≤κS≤κT. The isochoric heat capacity is
Stability implies well-defined positive standard deviations of the thermodynamic variables. Temperature fluctuations are determined by the isochoric specific heat,(β,N,V) representation thereof. Relative fluctuations such as scale with N−1/2. The variance of the chemical potential can be assembled as α in (2.13), U in (2.16) and CV in (2.19) substituted. The variances of the extensive variables (internal energy, particle count and entropy) will be derived in phase space, cf. Section 3.2.
3. Power-law ensembles in phase space
3.1. Classical probability density
We give a probabilistic interpretation of the classical spectral number density dρB(γ) in (2.9). To this end, we consider the phase space of n particles with measured3qi integration is over a volume V, and the dγi integrations range in the interval γcut≤γi≤∞, cf. after (2.3). The angular integration over the solid angle dΩ has been carried out and gives the factor 4π. The scale factor (2πħ)3n renders the measure dimensionless. The probability density on the n-particle states factorizes as 1/n! accounts for indistinguishability and the positive integer s for spin multiplicity. The normalization factor ξ included in the weight bn(p,q) is found by summation over the n-particle subspaces, (3.3) factorize, (3.3), we obtain the normalization constant ξ=logZB as stated in (2.8).
3.2. Fluctuations and entropy in phase space
The normalization condition (3.3) can be written as(3.3), ξ=logZB and the general identity ZB,β,α/ZB=ξ,β,α+ξ,βξ,α as well as the special properties ξ,α=−ξ and N=ξ(β,α,V). Thus, 〈n〉=N and 〈n2〉=N(N+1), keeping the volume V constant in contrast to the fluctuating particle number N.
Analogously, we find the energy averages. Since U(β,α,V)=−mξ,β, we can identify 〈u〉=U and 〈u2〉=−mU,β(β,α,V)+U2. Internal energy and particle count are correlated as 〈un〉=(1+N)U. The standard deviations are readily assembled with the expectation values , and (3.8). Defining Δn=n−〈n〉, etc., we find the covariance matrix elements as
3.3. Quantized power-law distributions
The partition function (2.6) is obtained via a trace calculation in fermionic occupation number representation,|n〉 for the occupation number representation of the fermionic creation and annihilation operators (labeled by the wave vector of the modes) can be found in Ref. . The particle number operator is , where . The energy operator is , where . The power-law weight factor hn(p,q) of the classical probability density in (3.2) is quantized as , , where . The energy operator is dimensionless, rescaled with the particle mass. is the Lorentz factor parameterized with the wave vector, and m stands for mc2. In the occupation number representation, the Hermitian number operators are diagonal, , and the fermionic occupation number attached to a wave vector can only take the values zero and one. (In the bosonic case sketched below, ranges between zero and infinity.) We also note the dispersion relation , relating electron energy and wave number . Finally, is the wave number of the lower energy threshold ħωcut=mγcut. Integration over wave vectors is defined by d3k=d3p/ħ3, with the momentum-space measure d3p parameterized by the Lorentz factor as in (3.1).
We employ box quantization, discretizing the wave vector as , so that summations are taken over integer lattice points in R3, corresponding to periodic boundary conditions on a box of size V=L3. By making use of , we can write trace (3.16) over the basis |n〉 asL→∞ amounts to replacing the summation over the lattice wave vectors by the integration , where the factor s accounts for spin degeneracy, γ as integration variable, , we arrive at the partition function logZF stated in (2.6).
We outline quantization of relativistic power-law ensembles in Bose–Einstein statistics ,  and . In this case, the summation over the occupation numbers in (3.17) extends to infinity,(3.18), (3.19), we arrive at the bosonic counterpart to the partition function (2.6), dρE(γ) can be identified via (2.3) and (3.22), dρF(γ) in (2.1) only by the minus sign in the denominator. s is a multiplicity index, and G(γ) is the spectral function (2.2). In the bosonic case, the reciprocal spectral function G−1(γ) must be bounded, its maximum being attained at γc. (γc can be infinite.) The fugacity z=e−α is restricted by z≤zc=G(γc), so that the denominator in (3.24) is positive. The classical regime is , whereas quantum condensation effects arise in the opposite limit, .
3.4. Hybrid quantum statistics
We briefly discuss hybrid distributions, in which the fermionic exclusion principle is softened to allow Nid identical particles labeled by the same wave vector . The partition function interpolating between Fermi and Bose statistics is found by extending the summation in (3.17),Nid. Summation of the finite geometric series in (3.25) gives logZH(Nid)<logZH(Nid+1). Performing the continuum limit, cf. (3.19), we find logZF in (2.6) and logZE in (3.22) are recovered for Nid=1 and Nid=∞, respectively. The spectral density dρH(Nid)(γ) of a hybrid quantum ensemble of fixed multiplicity Nid can be identified as in and via the particle count, Nid→∞ can be carried out in this representation of dρH(Nid)(γ), provided that βγ+α+logG(γ)>0, cf. after (3.24). Even if this constraint is not met, the positivity of density (3.29) is obvious in the equivalent representation Nid and the sign of exponent βγ+α+logG(γ), and dρH(Nid)(γ) stays finite when this exponent approaches zero. Alternatively, we may write the hybrid density (3.30) as 0<dρH(Nid)(γ)<dρH(Nid+1)(γ) is evident, cf. after (3.26), which holds for arbitrary real exponents βγ+α+logG(γ). In contrast to the Bose–Einstein limit Nid=∞, there is no constraint on the fugacity parameter α resulting in a condensed phase, so that the reciprocal spectral function 1/G(γ) need not be bounded in hybrid statistics, even for high multiplicity, so long as Nid is finite.
Remarks. To check the stability of the above quantum densities, one starts with the high- and low-temperature asymptotics of the thermodynamic variables, which can be done analytically in leading order  and . To check positivity of the specific heat and compressibility in the intermediate temperature regime, one has to specify the input parameters and study the cross-over numerically, in contrast to the classical variables discussed in Section 2.2. If positivity turns out to be violated for certain parameter ranges, the respective intervals have to be excluded. I also note that the power-law factor G(γ) drops out in the non-relativistic limit also in the quantum distributions, and has a pronounced effect only for large Lorentz factors, so that their applicability is currently limited to astrophysical plasmas. Whether the hybrid densities (3.29) can be related to real systems remains to be seen. The spectral fits in Section 5 are performed with the classical density (2.9).
4. Nonthermal spectral fitting in the ultra-relativistic regime: How to set it up in practice
We write the spectral number density (2.1) asγ=E/m, where m stands for mc2, and use dimensionless quantities: E and m in units of , where the factor 103d (usually with integer d) allows us to conveniently shift the energy scale. The units chosen for length, time and temperature are meter, second and Kelvin, so that and . Finally, β=m/(kBT), with and . The contribution of the Coulomb interaction ,  and  to the partition function is negligible in the case of a dilute ultra-relativistic plasma at high temperature, since e2/(4πrkBT)≪1, with interparticle distance r=(N/V)−1/3 and fine-structure constant e2/(4πħc)≈1/137 (Heaviside–Lorentz system), cf. Table 1.
logz e− 1.8×10−6 5.0×10−6 1.0×10 3 −96.8 −9.7×10 4 1.8×10−4 e+ 1.3×10−7 3.4×10−7 7.4×10 2 −95.3 −7.1×10 4 1.3×10−5
- Full-size table
We rescale the temperature and fugacity parameters, as well as the amplitudes in the power-law ratio g(γ) in (2.2),fB(E)=g−1(E/m) denotes the reparameterized reciprocal power-law ratio (2.2), (2.4), we define the scaling function as , where γmin is the smallest Lorentz factor admissible in the dispersion relation. The fermionic spectral number density (4.1) reads in dimensionless quantities dρB/dE is obtained by approximating the denominator by 1.
The spectral flux density is found by multiplying the number density (4.4) with the group velocity,dΩ is the solid angle increment and dρF/dE the particle density (4.4). Density F(E) is the spectral number flux; the spectral energy flux is obtained by rescaling F(E) with E. To avoid steep spectral slopes in log–log plots, which tend to conceal the fine-structure, it is convenient to rescale F(E) with a suitable power Ek, where k is usually integer. (We will use k=3 in Section 5.) In dimensionless quantities, cf. (4.4) and (4.6), (4.2), s is the spin multiplicity. The classical flux FB(E) (defined by the Boltzmann density dρB/dE substituted in (4.6)) is found by dropping the ξ0 term in the denominator of (4.7), 1−m2/E2 as well; this limit will be used in the spectral fits in Section 5.
In a wideband fit, it is convenient to represent the spectral function fB(E) in (4.9) by a series of power-law peaks ,and real exponents ηi,κi (determining location, height and the two power-law slopes of each peak) to be extracted from the spectral fit. Once the exponents and amplitudes on the right-hand side of (4.10) are known, the power-law ratio fB(E) in (4.3) is unambiguously defined. The amplitude and the exponent 2+k−δ can readily be identified as well, by considering the limit E→0 in (4.10).
Remark. The individual spectral components of nonthermal high-energy spectra are typically composed of a power-law ascent, followed by a slightly curved descending power-law slope due to an exponential cutoff  and . Power-law slopes are linear in log−log plots, clearly distinguishable from curved exponential cutoffs, and wideband spectra can readily be assembled with them. The power-law slopes are often blurred due to overlapping spectral peaks. The spectral function (4.10) models a sequence of such spectral peaks. Given that spectral fits are usually performed in double logarithmic representation, power-law ratios of type (2.2) and (4.3) seem to be sufficiently general, although more complicated spectral functions have been considered . The spectral functions (2.2) are constant for small Lorentz factors, g(γ→1)∼1, and have a power-law shape in the opposite limit, g(γ→∞)∝γσn−ρk. A different cross-over (other than in (2.2)) does not affect the thermodynamic stability, cf. Section 2.2, but if one replaces, for instance, the denominators in (4.10) by , it would not be possible to fit the electron spectrum in Fig. 1.
The chemical potential , temperature and fugacity z=e−α can be extracted from the fitting parameters and , cf. (4.2) and (4.8),ξ0≈3.18×10−9(d+6)(2/s), which becomes moderate on the 10−9 eV scale (d=−6), cf. (4.8) and after (4.1), and m≈0.511×10−3(d+1) for an electron/positron gas. The specific particle density N/V in (2.3) reads in dimensionless quantities (2.6), Ecut=mγcut. The energy density , cf. (2.5), is obtained by adding a factor E to the integrand in (4.12). (These integrals are numerically very tame.) The specific densities N/V and U/V become independent of the particle rest mass m in the ultra-relativistic regime , since the roots in (4.12) and (4.13) can be dropped, so that the mass is completely absorbed in the fitting parameters. The classical limit of the particle density N/V is found by putting the term in the denominator of the integrand in (4.12) to zero. Since ξ0 is extremely small on the GeV scale and the amplitude (extracted from the spectral fit) turns out to be moderate, we can safely assume and use the classical limit. Expanding the logarithm in the partition function (4.13) in leading order in , we recover N=logZB, cf. before (2.13). The entropy density (2.7) can be assembled from the classical specific number and energy densities in (4.12), α=−logz in (2.7). Fugacity and temperature are extracted from the fitting parameters by way of (4.11). As mentioned, in the ultra-relativistic limit, we do not need to know the mass of the particles to calculate the specific particle and energy densities (4.12) from the spectral fit. However, the particle mass affects the entropy density via the fugacity, cf. (4.11), as the entropy depends on the zero point of the chemical potential. This zero point is determined by the normalization of the spectral function G(γ) in (2.1) at γ=1. We have chosen G(1)=1, so that the chemical potential of the non-relativistic Fermi gas limit is recovered from the relativistic potential μ in (4.11) by subtraction of the rest mass, cf. after (2.10).
5. Spectral flux densities and thermodynamic parameters of cosmic-ray electrons/positrons
The spectral fits in Fig. 1 and Fig. 2 are performed with the classical ultra-relativistic flux density FB(E) in (4.9) and the spectral function fB(E) in (4.3). We employ a GeV energy scale (d=0, cf. after (4.1)), and scale the flux density with E3, cf. (4.9),(4.3) and (4.10), (5.1) are the power-law exponents δ,σ,ρ and amplitudes , as well as the exponent determining the cutoff. The ultra-relativistic limit applies, since the fit is done above a cutoff energy Ecut where m2/E2≪1. (We use Ecut≈1 on the GeV scale.) In this limit, the mass-square m2 is not a fitting parameter, as it is absorbed in the amplitudes . The classical spectral particle density reads, cf. (4.4) and (4.12), , cf. (4.2), (4.8) and (4.11). The spin multiplicity is s=2.
The electron flux Fe(E) in Fig. 1 is fitted with(5.1) and (5.2) can be read off from this equation. The positron fraction Rp(E)=Fp/(Fe+Fp), where Fp(E) denotes the positronic flux density, was measured by the AMS Collaboration , cf. Fig. 3, without disentangling the electron and positron flux, and an analytic fit of the positron fraction was derived, E is in GeV units. Combining the electronic flux density (5.4) with the AMS ratio (5.5), we find an analytic representation of the positron flux, Fig. 2, we approximate this analytic function by a power-law density Fp(E)
The goal is to find power-law densities suitable for ultra-relativistic wideband fitting. A consistent thermodynamic formalism requires these distributions to admit an extensive and stable entropy functional resulting in thermodynamic variables with positive standard deviations and relative fluctuations vanishing in the thermodynamic limit. Here, we have discussed a class of stable power-law densities, stationary non-equilibrium distributions defined by an empirical spectral function, a power-law ratio to be inferred from measured flux densities, cf. Sections 2 and 3.
We explained the spectral fitting with ultra-relativistic power-law densities, cf. Section 4, and applied it to cosmic-ray electrons treated as dilute nonthermal plasma, cf. Section 5. The cosmic-ray electron flux as well as the AMS and PAMELA positron fraction  and  are highly isotropic. The statistical description given here does not involve any hypothetical assumptions on isotropically distributed electron/positron sources and production mechanisms. The power-law densities are empirically determined from spectral fits , ,  and .
Cosmic-ray electrons/positrons constitute a dilute nonthermal two-component plasma, whose components can efficiently be modeled as classical ultra-relativistic power-law ensembles. The classical limit allows fairly explicit evaluation of the thermodynamic functions in (T,N,V) representation. High-temperature expansions are not efficient though, as they cannot adequately describe peak-to-peak cross-overs in wideband spectra. The electron and positron spectra in Fig. 1 and Fig. 2 are not wideband, extending from the low GeV to the low TeV region, but there are still two spectral peaks discernible, and the cross-over is beyond asymptotic expansions.
The cosmic-ray electron spectrum is well covered by several measurements in the GeV and low TeV range. Once the power-law fit of the spectral electron density is performed, cf. Fig. 1, it can be combined with the measured positron fraction, cf. Fig. 3, to obtain an estimate of the positronic flux density  and . The positron flux is modeled by a nonthermal power-law density composed of two spectral peaks like the electronic counterpart, cf. Fig. 2. The thermodynamic parameters of the electron/positron gas can be read off from the spectral fits. The temperature of the two non-equilibrated components is comparable but decidedly different, and their specific energy, entropy and number densities differ by roughly one order, cf. Table 1.
Astrophys. J., 413 (1993), p. 281
Astrophys. J. Suppl., 126 (2000), p. 19
Astrophys. J., 543 (2000), p. 722
Physica A, 391 (2012), p. 6690
Physica A, 392 (2013), p. 4388
Phys. Rev. Lett., 106 (2011), p. 201101
Phys. Rev. Lett., 108 (2012), p. 011103
Phys. Rev. D, 82 (2010), p. 092004
Physica A, 392 (2013), p. 982
Astron. Astrophys., 508 (2009), p. 561
Phys. Rev. Lett., 101 (2008), p. 261104
Physica A, 385 (2007), p. 558
Physica A, 387 (2008), p. 3480
Physica B, 405 (2010), p. 1022
Phys. Lett. A, 372 (2008), p. 4344
Phys. Lett. A (2013) http://dx.doi.org/10.1016/j.physleta.2013.10.016
Physica A, 307 (2002), p. 375
Physica A, 391 (2012), p. 2313
Physica A, 392 (2013), p. 3984
J. Chem. Phys., 45 (1966), p. 2102
Rev. Mod. Phys., 54 (1982), p. 1017
Phys. Rep., 149 (1987), p. 91
Phys. Rev. Lett., 110 (2013), p. 141102
Phys. Rev. Lett., 111 (2013), p. 081102
Astropart. Phys., 27 (2007), p. 92
EPL, 89 (2010), p. 39002
Opt. Commun., 282 (2009), p. 1710
Physica B, 404 (2009), p. 1383
Astrophys. J., 559 (2001), p. 296
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