Volume 405, Issue 3, 1 February 2010, Pages 1022-1032
Roman Tomaschitz^{}^{, }^{a}^{, }^{}
Abstract
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
Article Outline
- 1. Introduction
- 2. Ultra-relativistic Fermi power laws
- 3. Thermodynamic variables of nearly degenerate power-law distributions
- 4. Tachyonic cascade spectra
- 5. Conclusion
- Acknowledgements
- 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
- References
1. Introduction
Electronic 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 [1]. 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. [2]. 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 m_{t} is the mass of the superluminal Proca field A_{μ}, and q the tachyonic charge carried by the subluminal electron current [3]. 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}/∂t^{2}, 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 [4], [5], [6] and [7]. Estimates of the tachyon–electron mass ratio and the tachyonic fine structure constant are m_{t}/m≈1/238 and q^{2}/(4πc)≈1.0×10^{-13}, obtained from hydrogenic Lamb shifts [8].
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 [9], [10], [11], [12], [13] and [14]. 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 [15], [16], [17], [18] and [19]. 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 [20].
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. [21]. Here, we consider ultra-relativistic multi-component plasmas in the collisionless regime [22], in stationary non-equilibrium described by power-law densities [1], [23] and [24].
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 [25] and to a burst spectrum [26], 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
where the electronic Lorentz factors range in an interval γ_{1}≤γ<∞. γ_{1} is the lower edge of Lorentz factors of the electron distribution, the threshold energy being mγ_{1}, γ_{1}≥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 γ_{1}=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
Integral representation (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 [27]. The opposite asymptotic limit, the nearly degenerate quantum regime at low temperature and high density, is based on the representation
obtained from Eq. (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 γ_{1}=1. (The case γ_{1}>1 will be studied in Section 4.)
The μ→∞
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),
and the expansion of the partition function follows from Eq. (A.38):
For these asymptotics to be applicable, conditions m/μ1 and m/(βμ)1 have to be met. The identities
can be used to check the consistency of series expansions (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
as p_{F}:=(3π^{2}N/V)^{1/3},
we invert Eq. (2.6) in
ascending powers of 1/p_{F}:
We substitute μ(p_{F},β) into the internal energy (2.7) and partition function (2.8) to find
The mean energy per particle in the ultra-relativistic regime is thus U/N3p_{F}/4. The entropy is calculated via
where we used . On substituting series (3.1), (3.3) and (3.4), we obtain
The expansion parameter is x=m/(βp_{F}) or
The isochoric heat capacity reads
Thermodynamic stability requires C_{V}≥0, which is evidently satisfied. The Helmholtz free energy is assembled as
where we substitute the fugacity expansions of the chemical potential (3.1) and the partition function (3.4) as well as to obtain
with x=m/(βp_{F}) as in Eq. (3.7). The thermal equation of state
is thus found as
We solve this equation iteratively for p_{F}(β,P):
where y:=(12π^{2}P)^{1/4}. The thermal equation (3.12) can thus be written as
These expansions in ascending powers of 1/y are applicable if m/y1 as well as m/(βy)1. The isothermal compressibility reads
where, cf. Eq. (3.13),
so that we arrive at
with y=(12π^{2}P)^{1/4}. Thermodynamic stability requires κ_{T}≥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 β_{F}:=m/(kT_{F}) is not accessible with the fugacity expansions derived here, which require both β_{F}1 and β_{F}/β1. If β1/β_{F}, 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 [28]. 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 [29].
We have put =c=1.
To restore the dimensions, we rescale β=mc^{2}/(kT)
and p_{F}=(3π^{2}N/V)^{1/3}
so that kT_{F}μcp_{F},
and note .
The expansion parameter x=mc/(βp_{F})
is chosen to be dimensionless, and the dimension of y=(12π^{2}^{3}c^{3}P)^{1/4}
to be that of energy. The expansion parameter mc^{2}/y
in Eq. (3.13) is thus
dimensionless as well, and p_{F}y/c.
Hence
4. Tachyonic cascade spectra
The spectral averaging of tachyonic radiation densities with
electronic power-law distributions (2.2) has
already been explained in Ref. [30], 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:
The weight factors f_{k=1,2,3} denote the averages
where A_{F}:=m^{3}V/π^{2} is the normalization factor of the power-law density dρ_{F} in Eq. (2.2). The superscripts T and L in Eq. (4.2) refer to the transversal and longitudinal polarization components defined by and Δ^{L}=0 [31]. γ is the electronic Lorentz factor, α_{q} the tachyonic fine structure constant, and m_{t} 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:
The threshold Lorentz factor γ_{1} enters as lower integration boundary in the weights (4.3), and γ_{1}≥μ_{t}. In the thermodynamic variables, we can still put γ_{1}=1, cf. Eqs. (2.3), (2.4) and (2.5), but electrons with Lorentz factors below μ_{t} cannot radiate tachyonic quanta [32]. γ_{1} determines the break frequency
which enters in the step functions θ in Eq. (4.1), separating the spectrum into a low- and a high-frequency band. In particular, , and the smallest possible threshold, γ_{1}=μ_{t}, corresponds to ω_{1}=0. The threshold Lorentz factor μ_{t} 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 α_{q}=q^{2}/(4πc)≈1.0×10^{-13}. The tachyon mass is , and the tachyon–electron mass ratio m_{t}/m≈1/238. These estimates are obtained from hydrogenic Lamb shifts [8]. The particle number reads , where γ_{1} 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=mc^{2}/(kT) and the chemical potential by μ=-mα/β. The normalization factor A_{F} 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.
The weight factors (4.3) are
related to the Fermi integral F(a,
b) in Eq. (A.1) (with a=k-1
and b=0) and the Sommerfeld decomposition (A.3) by
The expansion parameter z is defined in Eq. (A.20):
where μ is the chemical potential, cf. after Eq. (4.5). For the asymptotic series (4.7) to be applicable, conditions βγ_{1}z1 as well as z>1 have to be satisfied. [Since b=0, we do not need to require γ_{1}z1, even though series (4.7) is a systematic ascending 1/z expansion, cf. Eq. (A.23) and after Eq. (A.34). In fact, F_{0} is a polynomial in z, and the factor ρ(γ_{1}z) in Eq. (A.21) drops out in F_{1}+F_{2} at b=0, cf. Eqs. (A.13) and (A.15), so that an additional temperature independent expansion in inverse powers of γ_{1}z is not needed if b=0 in the Fermi integral (A.1).]
The amplitude A_{F} and the fugacity e^{-α} are two independent fitting parameters in the source density dρ_{F}(γ), cf. Eqs. (2.2) and (4.3). This is in contrast to the classical limit, , where the factors of the amplitude A_{F}e^{-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 A_{F}e^{-α}/2, cf. Eq. (3.7) in Ref. [24]. In the ultra-relativistic limit, γ1, the factor γ^{-δ} can formally be generated by analytic continuation in the momentum space dimension, since pmγ [33]. 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 f_{k}(γ_{1})
with the chemical potential via z(μ)
in Eq. (4.8). The
independent fitting parameters in f_{k}(γ_{1})
are thus temperature β, threshold Lorentz factor γ_{1},
chemical potential μ, and the volume factor A_{F}
of the Fermi distribution in Eq. (4.3). We may
fix γ_{1}
at the lowest possible threshold, γ_{1}=μ_{t},
cf. Eq. (4.4) (and put γ_{1}=1
in the thermodynamic functions (2.3), (2.4) and (2.5)), so that
the averaged spectral densities (4.1) simplify to
which allows to expand the denominator in Eq. (2.2) to arrive in leading order at the classical power-law density, cf. after Eq. (4.8),
Here, we use the customary definition of the electronic power-law index, . (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 n^{e} obtained from the spectral fit, cf. after Eq. (4.15). The cascades ρ_{i=1,2} depicted in Fig. 1 and Fig. 2 are generated with and γ_{1}=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 .
Fig. 1. Spectral map of the Markarian blazar Mkn 421 in a low emission state. MAGIC flux points from Ref. [25]. The solid line T+L depicts the unpolarized differential tachyon flux dN^{T+L}/dE, obtained by adding the flux densities ρ_{1,2} of two ultra-relativistic electron populations, cf. Table 1, and rescaled with E^{2} for better visibility of the spectral curvature, cf. Eq. (4.13). The transversal and longitudinal flux densities dN^{T,L}/dE add up to the total flux T+L=ρ_{1}+ρ_{2}. The exponential decay of the cascades ρ_{1,2} sets in at about E_{cut}≈(m_{t}/m)kT, where m_{t}/m≈1/238 is the tachyon–electron mass ratio [8], implying cutoff energies of 550 GeV for the ρ_{1} cascade and 100 GeV for ρ_{2}. The spectral curvature resembles that of the TeV blazar W Comae in Fig. 2 of Ref. [27] at 450 Mpc, which suggests the curvature to be intrinsic rather than due to intergalactic absorption. One may also compare this cascade spectrum to the spectral maps of the BL Lacs 1ES 0347–212 at 830 Mpc, cf. Fig. 2 of Ref. [30], and 1ES 1218+304 at 800 Mpc, cf. Fig. 2 of Ref. [3]. There is no correlation of redshift and spectral curvature visible.
Thermal electronic source distributions ρ_{i} generating the tachyonic γ-ray cascades of the Markarian galaxy Mkn 421.
Markarian 421 z≈0.031, 140 Mpc | β | n^{e} | kT (TeV) | U (10^{60} erg) | |
---|---|---|---|---|---|
MAGIC, November 2004–April 2005 | |||||
ρ_{1} | 3.91×10^{−9} | 3.3×10^{−3} | 3.5×10^{57} | 131 | 2.2 |
ρ_{2} | 2.15×10^{−8} | 5.3×10^{−3} | 5.6×10^{57} | 23.8 | 0.64 |
Whipple, April 2004 | |||||
ρ_{1} | 8.96×10^{−10} | 8.5×10^{−3} | 9.0×10^{57} | 570 | 25 |
ρ_{2} | 9.77×10^{−9} | 2.5×10^{−2} | 2.7×10^{58} | 52.3 | 6.7 |
Each ρ_{i} stands for a Maxwell–Boltzmann density as defined in Eq. (4.11) (with δ=0 and γ_{1}=1). β is the cutoff parameter in the Boltzmann factor, and determines the amplitude of the tachyon flux generated by density ρ_{i}, 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 χ^{2} fit T+L=ρ_{1}+ρ_{2} in Fig. 1 and Fig. 2. The tachyonic cascades labeled ρ_{1,2} in the figures are produced by the corresponding electron densities listed in this table.
The classical limit of the fermionic spectral functions F^{T,L}(ω,γ_{1})
is the Boltzmann average B^{T,L}(ω,γ_{1}),
obtained by dropping all terms containing m_{t}/m
factors in Eq. (4.2), and
replacing dρ_{F}
in the spectral weights (4.3) by the
classical density
in Eq. (4.11). The
classical spectral weights reduce to incomplete gamma functions. Terms
containing m_{t}/m
factors in Eqs. (4.4) and (4.5) are
dropped as well, and the polarization coefficients reduce to
and ,
cf. after Eq. (4.3). The
classical limit of the averaged spectral densities p^{T,L}(ω)_{F}
in Eq. (4.1) thus reads
where the argument in the step functions is the classical limit of the break frequency (4.5).
The spectral fits of the BL Lacertae object (BL Lac) Markarian
421 in Fig. 1 and Fig. 2 are based
on the E^{2}-rescaled
flux densities [34]
we use a rescaled parameter for the fit,
which is independent of the distance estimate in Eq. (4.13). Here, implies the tachyon mass in keV units in the spectral functions (4.12). At γ-ray energies, only a tiny α_{q}/α_{e} 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 n_{1}, so that the actual number of radiating electrons is n^{e}:=n_{1}α_{e}/α_{q}≈7.3×10^{10}n_{1}. We thus find the electron count as , where defines the tachyonic flux amplitude extracted from the spectral fit [3]. This renormalized count n^{e} 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 [35]. The distance in Eq. (4.13) is inferred from the redshift via dcz/H_{0}, with , that is, h_{0}≈0.68. Hence, , and . The distance estimate does not affect the spectral maps, but the electron number n^{e}.
Fig. 1 shows the cascade fit of the Markarian galaxy Mkn 421 in a quiescent state [25], and Fig. 2 in a high emission state [26]. 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.
5. Conclusion
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 [3]. 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 [36].
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. [2].
Superluminal radiation from ultra-relativistic electrons orbiting in magnetic fields was investigated in Ref. [37]. 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. [37]. Thus we can use uniform radiation densities even in the presence of magnetic fields in the galactic nuclei.
Acknowledgments
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.
References
Appendix A. Incomplete integrals of fermionic power-law densities
The thermodynamic variables in Eqs. (2.3), (2.4) and (2.5) are
composed of integrals of type
We perform the substitution γ=γ_{1}(1+y)
in Eq. (A.1),
and use the Sommerfeld decomposition [39]:
The asymptotics of the temperature-dependent contribution F_{1}+F_{2} is discussed below, and the zero-temperature degeneracy F_{0} in Section A.2.
A.1. Fugacity expansion at low temperature and high density
The ascending
expansion of F_{1}+F_{2}
in Eq. (A.3) is found
by means of the substitutions y=λ(1-t)
in F_{1}
and y=λ(1+t)
in F_{2}.
Expanding the denominator in powers of ,
we obtain
where f(y) is defined in Eq. (A.2). In F_{1}, 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 g_{k}(t,δ,λ) at t=0:
and does not depend on the upper integration boundary if terms of O(1/Λ) are neglected [40]. 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
To obtain the Taylor coefficients a_{k,n}(δ,λ) in Eq. (A.7), we factorize g_{k}(t,δ,λ)=h_{1}h_{2}, cf. Eqs. (A.2) and (A.6),
and expand both factors. The Taylor series of h_{1} reads
As for h_{2}, we introduce the shortcuts
and find, by means of the Gegenbauer expansion (B.17):
We replace the binomial coefficient (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 g_{k}(t,δ,λ) in Eq. (A.7) as
The coefficients S(n,m;a) are calculated in Eqs. (B.15) and (B.16), and we put α_{n,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 F_{1,2} 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
The expansion of F_{2} is obtained from F_{1} by changing the sign of δ and inserting the factor (-1)^{n+1} into the n summation. The expansion of F_{1}+F_{2} is thus
where Δ_{n} are polynomials in as defined in Eqs. (B.11) and (B.12), and the coefficients α_{n,m}(λ) are defined by the finite series (A.15). By making use of (B.16) and (B.19), we calculate α_{n,m}(λ) for n=0,1,2:
The first three orders of the asymptotic expansion of F_{1}+F_{2} in Eq. (A.17) thus read
where Δ_{k=1,2,3} stands for Δ_{k}((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):
where Λ_{0} 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
We also note, cf. Eq. (B.12), , and
Expansion (A.19) is in ascending powers of 1/(βγ_{1}z), and holds for arbitrary exponents a,b, and γ_{1}≥1, provided that βγ_{1}z1. If in addition γ_{1}z1, we can also expand ρ in Eq. (A.21) to find
For this expansion to be valid, βγ_{1}z1 as well as γ_{1}z1 is required. That is, both conditions have to be met in a systematic 1/z expansion, where z is related to the fugacity Λ_{0}=e^{-α} as stated in Eq. (A.20).
A.2. Zero-temperature degeneracy
In the zero-temperature limit, the contribution F_{1}+F_{2}
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
substitute
into F_{0}
in Eq. (A.3), and
rescale t to find
We consider the limit λ→∞, and expand F_{0} 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 :
In the integrand of , we expand both factors in ascending powers of 1/t, and use term-by-term integration:
where (a)_{n} denotes the falling factorial, a(a-1)…(a-n+1). The ascending expansion of F_{0} 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).
We consider the special case γ_{1}=1,
that is υ_{1}=0
in Eq. (A.24). Here, b>-2
is necessary for integral (A.24) to
converge,
where ψ is the logarithmic derivative of the gamma function. The asymptotic expansion of the Fermi integral in Eqs. (A.1) and (A.3) is assembled with F_{1}+F_{2} 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 γ_{1}=1, we can use F_{0} in Eq. (A.29) or (A.30).
We list the integrals F_{0}(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),
where 0<arcsin<π/2, z>1, and γ_{1}≥1. The partition function (2.5) is assembled from
and there is also a zero-temperature contribution from F_{1}+F_{2} via βF(0,3), cf. Eq. (A.38). The spectral functions (4.10) are compiled at b=0, where
which is independent of γ_{1}, cf. Eq. (A.24).
We consider γ_{1}=1
and expand Eqs. (A.31) and (A.32) in 1/z
to find
Here, γ_{1}=1 is implied, but γ_{1} 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/(γ_{1}z). Eqs. (A.31), (A.32) and (A.33) give the fully degenerate contribution F_{0}(a,b) to the Fermi integral (A.1), (A.2) and (A.3). The fugacity expansion (A.34) of F_{0} 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 regime
The fugacity expansion of the Fermi integral F(a,b,γ_{1}=1)
in Eq. (A.1) is
obtained by adding F_{1}+F_{2}
in Eq. (A.23) (with γ_{1}=1)
and F_{0}
in Eq. (A.34). We find,
for exponents a=b=1:
with 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,γ_{1}=1) in Eq. (A.1) apply for and z1. The parameter Λ_{0} 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 γ_{1}=1, we can identify z=μ/m, cf. Eq. (A.20).
The fugacity expansions of the particle number and the
internal energy read, cf. Eqs. (2.3), (2.4) and (A.1),
Here, 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 [40], [41], [42], [43], [44], [45], [46] and [47]:
The first integral converges in the half-plane 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)Li_{s}(-z).
We note Li_{s}(-1)=(2^{1-s}-1)ζ(s);
in particular,
and Li_{2}(-1)=-π^{2}/12.
We will mainly consider real negative z or at least
z<1.
Li_{s}(z)
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 Li_{1}(z)=-log(1-z)
according to
where
is the Hurwitz zeta function, and r>0 is implied in Eq. (B.4). ζ(s,z) admits the integral representation [48]:
from which the large-z asymptotics is obtained by substituting the generating series of the Bernoulli numbers:
and using term-by-term integration (which amounts to applying Fourier asymptotics or Watson's lemma if the integration path is rotated into the imaginary axis [40]). The r→∞ asymptotics of the inversion formula (B.4) is thus found as [47]:
where
Here, (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
dropped, e^{iπs}Li_{s}(-1/r)=O(1/r),
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 [42]
Apparently Δ_{n}(1/r)=(-1)^{n}Δ_{n}(r). Finally, multiple s differentiation of Γ(s)Li_{s}(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:
where k, δ, and a are arbitrary real numbers. In particular, S(n,m;0)=S_{n,m}/n!, and we define S(n,m;a)=0 if m>n. For n=0,1,2, coefficients (B.15) read
In Eq. (A.13), we use an expansion in Gegenbauer polynomials, based on the generating series [48]:
Splitting the left-hand side into two factors (1-t/ρ_{±})^{α}, , we see that the expansion is absolutely convergent for |t|<min(|ρ_{±}|) and arbitrary ρ. More explicitly
where (λ)^{(n)} is the Pochhammer symbol Γ(λ+n)/Γ(λ). We note
Gegenbauer polynomials are a special class of hypergeometric polynomials,
At λ=1/2, they coincide with the Legendre polynomials P_{n}(x), and at λ=1 with the Chebyshev polynomials of the second kind U_{n}(x) [48]. Finally, , , and .