Roman Tomaschitz, a,
Polylogarithmic fugacity expansions of the partition function, the caloric and thermal equations of state, and the specific heat of fermionic power-law distributions are derived in the nearly degenerate low-temperature/high-density quantum regime. The spectral functions of an ultra-relativistic electron plasma are obtained by averaging the tachyonic radiation densities of inertial electrons with Fermi power-laws, whose entropy is shown to be extensive and stable. The averaged radiation densities are put to test by performing tachyonic cascade fits to the γ-ray spectrum of the TeV blazar Markarian 421 in a low and high emission state. Estimates of the thermal electron plasma in this active galactic nucleus are extracted from the spectral fits, such as temperature, number count, and internal energy. The tachyonic cascades reproduce the quiescent as well as a burst spectrum of the blazar obtained with imaging atmospheric Cherenkov detectors. Double-logarithmic plots of the differential tachyon flux exhibit intrinsic spectral curvature, caused by the Boltzmann factor of the electron gas.
Keywords: Fermi power-law ensembles; Nearly degenerate electron plasma; Superluminal radiation; Tachyonic cascade spectra; Spectral curvature; γ-Ray blazars; Negative mass-square; Transversal and longitudinal radiation modes; Polylogarithms
PACS classification codes: 52.25.Kn; 52.27.Ny; 71.10.Ca; 95.30.Tg
- 1. Introduction
- 2. Ultra-relativistic Fermi power laws
- 3. Thermodynamic variables of nearly degenerate power-law distributions
- 4. Tachyonic cascade spectra
- 5. Conclusion
- Appendix A. Incomplete integrals of fermionic power-law densities
- A.1. Fugacity expansion at low temperature and high density
- A.2. Zero-temperature degeneracy
- A.3. Number count, internal energy, and partition function in the nearly degenerate regime
- Appendix B. Polylogarithms, Stirling numbers, and Gegenbauer polynomials
1. IntroductionElectronic power-law distributions are commonly used in electromagnetic spectral averages to model the synchrotron emission of astrophysical plasmas, such as the magnetospheric X-ray emission of planets . In this article, we discuss spectral fitting based on fermionic power-law distributions, and develop the thermodynamic formalism of power-law ensembles quantized in Fermi–Dirac statistics. The quasiclassical fugacity expansion pertinent to fermionic power-law distributions in the high-temperature/low-density regime was derived in Ref. . Here, we investigate the opposite asymptotic limit, nearly degenerate ultra-relativistic power-law ensembles in the low-temperature/high-density quantum regime. The efficiency of the spectral averages is demonstrated by applying tachyonic cascade fits to the TeV blazar Mkn 421 in a low emission state and in outburst. The cascade spectra are obtained by averaging the tachyonic radiation densities of individual electrons over ultra-relativistic electron populations in the galactic nucleus. The thermodynamic parameters of the electron plasma are extracted from the spectral fits.
The tachyonic radiation field is a real Proca field with negative mass-square, , subject to the Lorentz condition , where mt is the mass of the superluminal Proca field Aμ, and q the tachyonic charge carried by the subluminal electron current . In the Proca equation, the mass term is added with a positive sign, and the sign convention for the d’Alembertian is ∂ν∂ν=Δ-∂2/∂t2, so that is the negative mass-square of the radiation field. The negative mass-square refers to the radiation field rather than the current, in contrast to traditional theories based on superluminal source particles emitting electromagnetic radiation , ,  and . Estimates of the tachyon–electron mass ratio and the tachyonic fine structure constant are mt/m≈1/238 and q2/(4πc)≈1.0×10-13, obtained from hydrogenic Lamb shifts .
Tachyonic radiation implies superluminal signal transfer, the energy quanta propagating faster than light in vacuum, due to their negative mass-square, in contrast to rotating superluminal light sources emitting vacuum Cherenkov radiation , , , ,  and . This superluminal energy propagation by tachyonic vacuum modes is also to be distinguished from superluminal group velocities arising in photonic crystals, optical fibers, or metamaterials , , ,  and . In contrast to tachyonic quanta, the actual signal speed defined by the electromagnetic energy flow in these media is always subluminal and occasionally even opposite to the group velocity .
In Section 2, we derive the fugacity expansion of the internal energy and the partition function in the low-temperature/high-density regime. The thermodynamic variables of nearly degenerate power-law distributions are calculated in Section 3, such as the thermal equation of state, entropy, and free energy. We check the positivity of the isochoric heat capacity and the isothermal compressibility for arbitrary power-law index, demonstrating thermodynamic stability in the quantum regime. A generalization of non-relativistic Fermi–Dirac distributions by way of modified dispersion relations has also been suggested in Ref. . Here, we consider ultra-relativistic multi-component plasmas in the collisionless regime , in stationary non-equilibrium described by power-law densities ,  and .
In Section 4, we study tachyonic radiation densities averaged over electronic power-law distributions, calculate the spectral functions in the nearly degenerate quantum regime, and discuss the range of applicability of the fugacity expansion, including the crossover into the quasiclassical regime. We perform cascade fits to the γ-ray flux of the Markarian galaxy Mkn 421 (at redshift z≈0.031) in a quiescent state  and to a burst spectrum , and compare with the tachyonic spectral maps of other BL Lacertae objects. The spectral curvature is reproduced by the tachyonic cascades without resort to intergalactic attenuation mechanisms.
In Section 5, we present our conclusions. In Appendix A, the Sommerfeld asymptotics of incomplete Fermi integrals is derived, which is the basis of the fugacity expansion of the thermodynamic functions in Section 3 and the spectral functions in Section 4. This involves polylogarithms discussed in Appendix B.
2. Ultra-relativistic Fermi power laws
We start with the partition function
γ1≤γ<∞. γ1 is the lower edge of Lorentz factors of the electron distribution, the threshold energy being mγ1, γ1≥1. The fugacity exponent α is related to the chemical potential by μ=-mα/β. δ is the electronic power-law exponent, and β=m/(kT) the cutoff parameter in the Boltzmann factor, so that the Fermi–Dirac equilibrium distribution is recovered with δ=0 and γ1=1. Here, we study non-thermal power-law ensembles of arbitrary real power-law index δ. The grand partition function (2.1) is obtained via a standard trace calculation in fermionic occupation number representation. Internal energy and particle number read
(2.1) of the partition function Z(δ,β,α,V) is the starting point for the quasiclassical fugacity expansion of the thermodynamic functions, applicable at high temperature and low density . The opposite asymptotic limit, the nearly degenerate quantum regime at low temperature and high density, is based on the representation
(2.1) by partial integration. We replace α by the chemical potential μ=-mα/β, and consider U, N, and as functions of the independent variables μ and β. From now on, we put γ1=1. (The case γ1>1 will be studied in Section 4.)
asymptotics of the above variables is assembled from the fugacity
expansion of the Fermi integrals in Appendix A. The
ascending 1/μ series of the particle number reads,
cf. Eqs. (A.35) and (A.37),
m/μ1 and m/(βμ)1 have to be met. The identities
(2.6), (2.7) and (2.8).
3. Thermodynamic variables of nearly degenerate power-law distributions
The thermodynamic functions are obtained by iteratively
solving (2.6) for the
chemical potential μ, which is then substituted
into the asymptotic series (2.7) and (2.8) of the
internal energy and the partition function. Defining the Fermi momentum
we invert Eq. (2.6) in
ascending powers of 1/pF:
μ(pF,β) into the internal energy (2.7) and partition function (2.8) to find
U/N3pF/4. The entropy is calculated via
. On substituting series (3.1), (3.3) and (3.4), we obtain
CV≥0, which is evidently satisfied. The Helmholtz free energy is assembled as
(3.1) and the partition function (3.4) as well as to obtain
x=m/(βpF) as in Eq. (3.7). The thermal equation of state
y:=(12π2P)1/4. The thermal equation (3.12) can thus be written as
m/y1 as well as m/(βy)1. The isothermal compressibility reads
y=(12π2P)1/4. Thermodynamic stability requires κT≥0, which is satisfied.
In the fully degenerate case, at zero temperature, the power-law exponent δ drops out in all thermodynamic variables, and at finite β it does not enter in leading order. The regime below βF:=m/(kTF) is not accessible with the fugacity expansions derived here, which require both βF1 and βF/β1. If β1/βF, the first-order correction proportional to δ is overpowered by the second order, which is independent of δ in this limit. Therefore, the Chandrasekhar mass limit of white dwarfs is not affected by the power-law exponent, as it assumes total degeneracy . Order-of-magnitude estimates of cooling times derived from homology relations are not affected either, as they are based on the leading-order temperature and density scaling of pressure and specific heat .
We have put =c=1.
To restore the dimensions, we rescale β=mc2/(kT)
so that kTFμcpF,
and note .
The expansion parameter x=mc/(βpF)
is chosen to be dimensionless, and the dimension of y=(12π23c3P)1/4
to be that of energy. The expansion parameter mc2/y
in Eq. (3.13) is thus
dimensionless as well, and pFy/c.
4. Tachyonic cascade spectra
The spectral averaging of tachyonic radiation densities with
electronic power-law distributions (2.2) has
already been explained in Ref. , where we
mainly focused on the quasiclassical regime, but also derived the
general formalism applicable in the nearly degenerate case. In Eqs. (4.1), (4.2), (4.3), (4.4) and (4.5), we
summarize the averaged radiation densities:
fk=1,2,3 denote the averages
AF:=m3V/π2 is the normalization factor of the power-law density dρF in Eq. (2.2). The superscripts T and L in Eq. (4.2) refer to the transversal and longitudinal polarization components defined by and ΔL=0 . γ is the electronic Lorentz factor, αq the tachyonic fine structure constant, and mt the tachyon mass. The argument in the second spectral function in Eq. (4.1) is the minimal electronic Lorentz factor for radiation at this frequency:
γ1 enters as lower integration boundary in the weights (4.3), and γ1≥μt. In the thermodynamic variables, we can still put γ1=1, cf. Eqs. (2.3), (2.4) and (2.5), but electrons with Lorentz factors below μt cannot radiate tachyonic quanta . γ1 determines the break frequency
(4.1), separating the spectrum into a low- and a high-frequency band. In particular, , and the smallest possible threshold, γ1=μt, corresponds to ω1=0. The threshold Lorentz factor μt depends on the tachyon–electron mass ratio, cf. Eq. (4.4), and is not to be confused with the chemical potential μ.
The units =c=1 can easily be restored. We use the Heaviside–Lorentz system, so that αq=q2/(4πc)≈1.0×10-13. The tachyon mass is , and the tachyon–electron mass ratio mt/m≈1/238. These estimates are obtained from hydrogenic Lamb shifts . The particle number reads , where γ1 is the lower edge of Lorentz factors in the source population. The exponential cutoff in the spectral weights (4.3) is related to the electron temperature by β=mc2/(kT) and the chemical potential by μ=-mα/β. The normalization factor AF is dimensionless via m→mc/; the volume factor in the thermodynamic functions in Section 3 is thus found as , where is the reduced electronic Compton wavelength.AF as in Eq. (4.3). We substitute F0(k-1,0)=(zk-1)/k, cf. Eq. (A.33), and assemble the temperature dependent contribution F1+F2 in Eq. (4.6) by making use of Eqs. (A.19), (A.20), (A.21) and (A.22) (with a=k-1 and b=0),
(4.5). For the asymptotic series (4.7) to be applicable, conditions βγ1z1 as well as z>1 have to be satisfied. [Since b=0, we do not need to require γ1z1, even though series (4.7) is a systematic ascending 1/z expansion, cf. Eq. (A.23) and after Eq. (A.34). In fact, F0 is a polynomial in z, and the factor ρ(γ1z) in Eq. (A.21) drops out in F1+F2 at b=0, cf. Eqs. (A.13) and (A.15), so that an additional temperature independent expansion in inverse powers of γ1z is not needed if b=0 in the Fermi integral (A.1).]
The amplitude AF and the fugacity e-α are two independent fitting parameters in the source density dρF(γ), cf. Eqs. (2.2) and (4.3). This is in contrast to the classical limit, , where the factors of the amplitude AFe-α cannot independently be determined from the spectral fit. This amplitude differs from a classical Boltzmann power-law distribution due to the fermionic multiplicity factor, the Boltzmann normalization being AFe-α/2, cf. Eq. (3.7) in Ref. . In the ultra-relativistic limit, γ1, the factor γ-δ can formally be generated by analytic continuation in the momentum space dimension, since pmγ . In the case of a genuine fermionic power-law distribution in the nearly degenerate quantum regime, one can determine the volume as well as the particle number from the spectral fit, cf. after Eq. (4.5).
We parametrize the spectral weights fk(γ1)
with the chemical potential via z(μ)
in Eq. (4.8). The
independent fitting parameters in fk(γ1)
are thus temperature β, threshold Lorentz factor γ1,
chemical potential μ, and the volume factor AF
of the Fermi distribution in Eq. (4.3). We may
at the lowest possible threshold, γ1=μt,
cf. Eq. (4.4) (and put γ1=1
in the thermodynamic functions (2.3), (2.4) and (2.5)), so that
the averaged spectral densities (4.1) simplify to
(2.2) to arrive in leading order at the classical power-law density, cf. after Eq. (4.8),
. (The electron index is denoted by a hat, to avoid confusion with the parameter α in Eq. (4.10) defining the fugacity e-α and the chemical potential.) The normalization factor is related to the particle number via to be identified with the renormalized electron count ne obtained from the spectral fit, cf. after Eq. (4.15). The cascades ρi=1,2 depicted in Fig. 1 and Fig. 2 are generated with and γ1=1, that is, with Maxwell–Boltzmann distributions as specified in Table 1. Condition (4.10) can even be met at high temperature, at sufficiently high frequency, cf. Eq. (4.4), implying exponential decay of the spectral functions .
Thermal electronic source distributions ρi generating the tachyonic γ-ray cascades of the Markarian galaxy Mkn 421.
Each ρi stands for a Maxwell–Boltzmann density as defined in Eq. (4.11) (with δ=0 and γ1=1). β is the cutoff parameter in the Boltzmann factor, and determines the amplitude of the tachyon flux generated by density ρi, from which the electron count is inferred at the indicated distance. kT is the temperature and U the internal energy of the electron populations, cf. after Eq. (4.15). Each cascade depends on two fitting parameters β and extracted from the χ2 fit T+L=ρ1+ρ2 in Fig. 1 and Fig. 2. The tachyonic cascades labeled ρ1,2 in the figures are produced by the corresponding electron densities listed in this table.
The classical limit of the fermionic spectral functions FT,L(ω,γ1)
is the Boltzmann average BT,L(ω,γ1),
obtained by dropping all terms containing mt/m
factors in Eq. (4.2), and
in the spectral weights (4.3) by the
in Eq. (4.11). The
classical spectral weights reduce to incomplete gamma functions. Terms
factors in Eqs. (4.4) and (4.5) are
dropped as well, and the polarization coefficients reduce to
cf. after Eq. (4.3). The
classical limit of the averaged spectral densities pT,L(ω)F
in Eq. (4.1) thus reads
in the step functions is the classical limit of the break frequency (4.5). the spectral average (4.12). The cascade fits are performed with the unpolarized flux density dNT+L=dNT+dNL of thermal electron populations (4.11) (, γ1=1). Each electron density generates a cascade ρi, and the spectral map is obtained by adding two cascade spectra labeled ρ1,2 in the figures. As for the electron count
for the fit,
(4.13). Here, implies the tachyon mass in keV units in the spectral functions (4.12). At γ-ray energies, only a tiny αq/αe fraction (the ratio of tachyonic and electric fine structure constants) of the tachyon flux is absorbed by the detector, which requires a rescaling of the electron count n1, so that the actual number of radiating electrons is ne:=n1αe/αq≈7.3×1010n1. We thus find the electron count as , where defines the tachyonic flux amplitude extracted from the spectral fit . This renormalized count ne is to be identified with the particle number N in the thermodynamic variables. The electron temperature and cutoff parameter in the Boltzmann factor are related by , and the energy estimates in Table 1 are based on . The distance in Eq. (4.13) is inferred from the redshift via dcz/H0, with , that is, h0≈0.68. Hence, , and . The distance estimate does not affect the spectral maps, but the electron number ne.
Fig. 1 shows the cascade fit of the Markarian galaxy Mkn 421 in a quiescent state , and Fig. 2 in a high emission state . The redshift of Mkn 421 is z≈0.031, implying a distance of 140 Mpc. TeV γ-ray spectra of active galactic nuclei are usually assumed to be generated by inverse Compton scattering or pp scattering followed by pion decay. Both mechanisms result in a flux of TeV photons, assumed to be partially absorbed by interaction with background photons due to pair creation, so that the intrinsic spectrum has to be reconstructed on the basis of intergalactic absorption models depending on vaguely known cosmological input parameters. In contrast, the extragalactic tachyon flux is not attenuated by interaction with the background light, there is no absorption of tachyonic γ-rays. The curvature of the γ-ray spectra in double-logarithmic plots is caused by the Boltzmann factor of the electron densities generating the tachyon flux, so that the observed spectrum is already the intrinsic one, and no reconstruction is needed. The curvature present in the γ-ray spectra of BL Lacs is not correlated with distance; the spectral curvature does not increase with redshift if we compare the spectral fits in Fig. 1 and Fig. 2 to the spectral maps of other active galactic nuclei, cf. the figure captions.
Tachyonic γ-ray spectra of active galactic nuclei are generated by ultra-relativistic electron populations. Tachyons are radiation modes, unrelated to electromagnetic radiation. Electrons radiate tachyons, and these tachyonic quanta produce the observed γ-ray cascades. The tachyonic radiation modes are coupled by minimal substitution to the electron current. This field theory, a real Proca field with negative mass-square, admits a static potential analogous to the Coulomb potential, but oscillating because of the negative mass-square, and much weaker due to the small tachyonic fine structure constant. In the spectral maps, the tachyon–electron mass ratio shows in the cutoff energy of the cascades. The negative mass-square of tachyons implies superluminal velocity and allows longitudinal polarization. The tachyonic radiation field does not couple to electromagnetic fields, nor is it affected by electric charge. Thus, interaction of tachyons with photons can only happen indirectly via matter fields. In contrast to electromagnetic γ-rays, there is no extinction of the extragalactic tachyon flux by the cosmic background light, as tachyons do not interact with infrared photons. The ultra-relativistic electron plasma in the active galactic nucleus produces tachyonic γ-rays propagating unattenuated over intergalactic distances.
The tachyonic radiation density averaged over the electron populations in the galactic nucleus is generated by electrons in uniform motion. In particular, there is no electromagnetic radiation damping, as photons can only be radiated by accelerated charges, in contrast to tachyonic quanta, where the emission rate primarily depends on the electronic Lorentz factor rather than on acceleration . The high plasma temperature inferred from the spectral fits implies ultra-high energy electrons. Such high electron temperatures are also found in Galactic pulsar wind nebulae, production sites of ultra-high energy cosmic rays .
Specifically, we fitted a quiescent as well as a flare spectrum of the γ-ray blazar Mkn 421 with tachyonic cascades, and found that the spectral curvature is intrinsic and reproduced by the superluminal spectral densities (4.1) averaged with ultra-relativistic electron distributions. The curvature present in the γ-ray spectra of active galactic nuclei is not correlated with distance, so that absorption of electromagnetic radiation due to interaction with background photons is not a viable explanation for the spectral curvature. By contrast, there is no intergalactic attenuation of the tachyon flux, as tachyons cannot interact with photons. In Table 1, we have given estimates of the temperature, the source count, and the internal energy of the electron populations in the galactic nucleus generating the superluminal γ-ray cascades.
The fugacity expansion of the thermodynamic variables of a nearly degenerate electron plasma was derived in Sections 2 and 3, and fermionic power-law densities were shown to admit a stable and extensive entropy function in the quantum regime, cf. Eqs. (3.5) and (3.6). In Section 4, we averaged tachyonic spectral densities with electronic power-law distributions, and obtained the fugacity expansion of the quantized spectral functions in the nearly degenerate low-temperature/low-frequency/high-density regime. The integral representation (4.3) of the spectral weights covers the crossover into the quasiclassical high-temperature/high-frequency/low-density regime studied in Ref. .
Superluminal radiation from ultra-relativistic electrons orbiting in magnetic fields was investigated in Ref. . In the zero-magnetic-field limit, the averaged tachyonic synchrotron densities converge to the spectral densities (4.12). Orbital curvature induces modulations in the spectral slopes, but these ripples are attenuated when performing a pitch-angle average, cf. Figs. 1–3 of Ref. . Thus we can use uniform radiation densities even in the presence of magnetic fields in the galactic nuclei.
The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.
Appendix A. Incomplete integrals of fermionic power-law densitiesΛ0:=e-α denotes the fugacity. We derive the Λ0→∞ asymptotics of this integral, with lower integration boundary γ1≥1 and real parameters Λ0>0, β>0. The exponents a and b are moderate real numbers, and so is the power-law exponent δ. If the integral is complete, γ1=1, we have to require b>-2 for convergence.
F1+F2 is discussed below, and the zero-temperature degeneracy F0 in Section A.2.
A.1. Fugacity expansion at low temperature and high density
f(y) is defined in Eq. (A.2). In F1, we extend the upper integration boundary to infinity, which is justified by Watson's lemma, as the error is of order O(1/Λ), as compared to the expansion in powers of . We can replace the upper integration boundaries in Eqs. (A.4) and (A.5) by an arbitrary ε, since the asymptotic series is determined by the Taylor coefficients of gk(t,δ,λ) at t=0:
O(1/Λ) are neglected . For technical convenience, we extend the integration boundaries to infinity, even though series (A.7) may only have a small radius of convergence, cf. Eq. (B.17). (In this way, we avoid incomplete gamma functions in the term-by-term integrations in Eqs. (A.4) and (A.5), which would have to be expanded to arrive at series (A.8) and (A.9) below.) On substituting series (A.7) into Eqs. (A.4) and (A.5), and interchanging integration and summations, we find
ak,n(δ,λ) in Eq. (A.7), we factorize gk(t,δ,λ)=h1h2, cf. Eqs. (A.2) and (A.6),
h2, we introduce the shortcuts
(kδ+a)m/m! in Eq. (A.11) by the Stirling expansion (B.14), and use the product of series (A.11) and (A.13) to find the Taylor coefficients of gk(t,δ,λ) in Eq. (A.7) as
S(n,m;a) are calculated in Eqs. (B.15) and (B.16), and we put αn,m(λ)=0 for m>n. The Gegenbauer polynomials are listed in Eqs. (B.18), (B.19) and (B.20). We substitute the Taylor coefficients (A.14) into series F1,2 in Eqs. (A.8) and (A.9), and interchange the summations. In this way, the ascending series of the polylog (B.1) is recovered, so that
F2 is obtained from F1 by changing the sign of δ and inserting the factor (-1)n+1 into the n summation. The expansion of F1+F2 is thus
Δn are polynomials in as defined in Eqs. (B.11) and (B.12), and the coefficients αn,m(λ) are defined by the finite series (A.15). By making use of (B.16) and (B.19), we calculate αn,m(λ) for n=0,1,2:
F1+F2 in Eq. (A.17) thus read
Δk=1,2,3 stands for Δk((1+λ)δ), cf. Eqs. (B.11) and (B.12), and ρ is defined in Eq. (A.12). We introduce z=1+λ as expansion parameter (in ascending powers of 1/z):
Λ0 is the fugacity, cf. Eq. (A.1), and , cf. Eq. (A.2). λ is positive since Λ>1 is required in Eqs. (A.3), (A.4), (A.5) and (A.6). In Eq. (A.19), we substitute 1+λ=z as well as
(B.12), , and
(A.19) is in ascending powers of 1/(βγ1z), and holds for arbitrary exponents a,b, and γ1≥1, provided that βγ1z1. If in addition γ1z1, we can also expand ρ in Eq. (A.21) to find
βγ1z1 as well as γ1z1 is required. That is, both conditions have to be met in a systematic 1/z expansion, where z is related to the fugacity Λ0=e-α as stated in Eq. (A.20).
A.2. Zero-temperature degeneracy
In the zero-temperature limit, the contribution F1+F2
to the Fermi integral F(a,b)
in Eq. (A.1) vanishes,
cf. Eqs. (A.19) and (A.23), so that F
reduces to the temperature-independent residual ,
cf. Eq. (A.3). We
in Eq. (A.3), and
rescale t to find
λ→∞, and expand F0 in ascending powers of . To this end, we split the integral (A.24) into , and write , where I stands for the right-hand side of Eq. (A.24) with lower and upper integration boundaries as indicated. Apparently, scales out in :
, we expand both factors in ascending powers of 1/t, and use term-by-term integration:
(a)n denotes the falling factorial, a(a-1)…(a-n+1). The ascending expansion of F0 in Eq. (A.24) is thus obtained as , with series (A.26) and (A.27) substituted. In the case of integer exponents a and b, singularities can arise in the series coefficients of and , which cancel if ε expanded, cf. Eq. (A.30). (A.24) via the decomposition or from Eqs. (A.26), (A.27) and (A.28), the first term on the right-hand side in Eq. (A.29) being and the second . A singularity may occur in the first term due to a pole of the second gamma function in the nominator. A corresponding singularity arises in a series coefficient of the hypergeometric term, so that the singularities cancel if ε expanded . The poles occur at a=2k-1-b, k=0,1,2,…, for arbitrary real b>-2; we find F0(γ1=1,a=2k-1-b) as
in Eqs. (A.1) and (A.3) is assembled with F1+F2 in Eq. (A.17) and in Eqs. (A.26), (A.27) and (A.28). In the case that F is complete, with lower integration boundary γ1=1, we can use F0 in Eq. (A.29) or (A.30).
We list the integrals F0(a,b)
(defined in Eq. (A.24))
occurring in the thermodynamic functions (2.3), (2.4) and (2.5). As in Eq.
(A.20), we put z=1+λ,
so that ,
cf. Eq. (A.25), and
obtain by elementary integration of Eq. (A.24),
0<arcsin<π/2, z>1, and γ1≥1. The partition function (2.5) is assembled from
F1+F2 via βF(0,3), cf. Eq. (A.38). The spectral functions (4.10) are compiled at b=0, where
γ1, cf. Eq. (A.24).
γ1=1 is implied, but γ1 can readily be scaled into these series according to Eqs. (A.31) and (A.32), so that the expansions are in ascending powers of 1/(γ1z). Eqs. (A.31), (A.32) and (A.33) give the fully degenerate contribution F0(a,b) to the Fermi integral (A.1), (A.2) and (A.3). The fugacity expansion (A.34) of F0 is needed even though the exact result (A.31) and (A.32) is known, since we have to iteratively solve for z when calculating the thermodynamic variables, cf. Section 3.
A.3. Number count, internal energy, and partition function in the nearly degenerate regimeb=1 in the 1/z2 and 1/z3 terms. The expansion of F(-1,3) is likewise given by Eq. (A.35) with b=3, and
b=1. The latter is also the expansion of F(0,3), if we put b=3 in the third and fourth term. The ellipses in Eqs. (A.35) and (A.36) stand for terms of . Expansions (A.35) and (A.36) of the Fermi integral F(a,b,γ1=1) in Eq. (A.1) apply for and z1. The parameter Λ0 in Eq. (A.1) is identified with the fugacity e-α, which is related to the chemical potential by α=-βμ/m, where β=m/(kT). Hence, if γ1=1, we can identify z=μ/m, cf. Eq. (A.20). (A.35) and (A.36) substituted. The expansion of the partition function in Eq. (2.5) (γ1=1) is assembled as
F(0,3) is series (A.36), and F(-1,3) series (A.35), both with b=3. The ellipsis stands for terms of .
Appendix B. Polylogarithms, Stirling numbers, and Gegenbauer polynomials
To keep this article self-contained, we summarize some
technical concepts as well as the notation used in the expansion of the
Fermi integrals in Appendix A. We
start with the series representation of the polylogarithm , , , , , ,  and :
Re(s)>1, the second and third require Re(s)>0; otherwise they are identical via partial integration and obvious substitutions. Series (B.1) is recovered by expansion in ascending powers and term-by-term integration. Fermi–Dirac integrals are defined by the third representation in Eq. (B.2), as -Γ(s)Lis(-z).
We note Lis(-1)=(21-s-1)ζ(s);
We will mainly consider real negative z or at least
is analytic in the z plane with branch cut (1,∞)
along the positive real axis, except for s=0
and at negative integer s, where the polylogs are
rational functions, obtained by recursive differentiation of Li1(z)=-log(1-z)
r>0 is implied in Eq. (B.4). ζ(s,z) admits the integral representation :
). The r→∞ asymptotics of the inversion formula (B.4) is thus found as :
(s)n is the falling factorial s(s-1)…(s-n+1), not to be confused with the Pochhammer symbol or rising factorial (s)(n):=s(s+1)…(s+n-1), so that (s)n=(-1)n(-s)(n).
The second polylog on the left-hand side of Eq. (B.8) can be
as terms of this order have been discarded in the expansion procedure.
However, the asymptotic series terminates for integer s≥0,
and then Eq. (B.8) holds true
as an identity even for small r. In the case of
negative integer s, the right-hand side of Eq. (B.8) vanishes
due to the poles of Γ(1+s)
in the denominator. In Eq. (B.8), we put s=n=0,1,2,…
to find 
Δn(1/r)=(-1)nΔn(r). Finally, multiple s differentiation of Γ(s)Lis(z) amounts to adding a factor of to the integrand of the third (Fermi–Dirac) integral in Eq. (B.2). The series expansions of these integrals are obtained via term-by-term differentiation of the ascending series (B.1) and the asymptotic expansion (B.8).
In Appendix A, we
also need the polynomial expansion of the falling factorial:
S(n,m;0)=Sn,m/n!, and we define S(n,m;a)=0 if m>n. For n=0,1,2, coefficients (B.15) read
(A.13), we use an expansion in Gegenbauer polynomials, based on the generating series :
(1-t/ρ±)α, , we see that the expansion is absolutely convergent for |t|<min(|ρ±|) and arbitrary ρ. More explicitly
(λ)(n) is the Pochhammer symbol Γ(λ+n)/Γ(λ). We note
λ=1/2, they coincide with the Legendre polynomials Pn(x), and at λ=1 with the Chebyshev polynomials of the second kind Un(x) . Finally, , , and .