Modeling electrical resistivity and particle fluxes with multiply broken power-law distributions

Abstract

Multiply broken power-law densities are introduced to model empirical data sets extending over several logarithmic decades. Two examples demonstrating the wide range of applicability of these distributions are discussed. First, the temperature dependence of the electrical resistivity of metals (Cu, Au, Cr, Al, W, Fe, Ni) is inferred by nonlinear least-squares regression covering the solid phase up to the melting point. The regressed broken power laws are compared with high- and low-temperature scaling predictions obtained from electron–phonon and electron–electron scattering. In the intermediate temperature range, inflection points arise in Log–Log plots of the electrical resistivity, unaccounted for by the Bloch–Grüneisen theory. In the second example, the energy evolution of cosmic-ray electron and positron fluxes is analyzed in terms of multiply broken and exponentially cut power-law densities, based on recently obtained number counts. Index functions quantifying the spectral variation and survival functions (complementary cumulative distributions) of the particle and energy fluxes are calculated from the regressed densities.

Graphic abstract

Introduction

The aim is to demonstrate the efficiency of multiply broken power-law distributions in high-precision modeling of empirical data. Two very different examples will be analyzed to that effect, the temperature evolution of the electrical resistivity of metals and the energy evolution of particle fluxes, the latter specified by cosmic-ray electron and positron spectra.

The main part of this paper deals with the mentioned applications, but we will also sketch the general formalism of power-law densities [1,2,3,4,5,6], their logarithmic representation and Index functions, their complementary cumulative distribution functions (CCDFs), as well as Weibull [7,8,9,10,11,12,13] and lognormal [14,15,16,17,18,19] cutoffs. The nonlinear least-squares regression of multiply broken and exponentially cut power-law densities will be discussed as well.

Multiply broken power-laws are introduced in Sect. 2.1, composed of power-law segments with smooth transitions at the break points, which are particularly suitable for the modeling of data sets extending over several logarithmic decades [20, 21]. In Sect. 2.2, we study cutoff factors (which are asymptotically Weibull exponentials or lognormals) for multiply broken power-law distributions and derive their Index functions. In Sect. 22,23,24,25] are performed. We study the Index functions of the total and intrinsic resistivities and compare the regressed temperature evolution with the linear temperature scaling of the Drude resistivity/conductivity theory in the high-temperature regime and with the T2 and T5 low-temperature scaling predicted by electron–electron and electron–phonon scattering, cf. e.g., [26, 27]. (The intrinsic resistivity is obtained by subtraction of the constant residual resistivity caused by impurity scattering.) In most cases, power-law scaling laws with integer exponents do not give accurate descriptions of the high- and low-temperature limits of the empirical resistivity data, not even approximately.

In Sect. 4, the modeling of cosmic-ray particle fluxes with multiply broken power-law densities is studied. The recently observed electron and positron fluxes [28,29,30] can accurately be described by power-law distributions with two spectral breaks and an exponential cutoff. At first sight, the decaying flux densities seem to admit simple power-law tails over an extended energy range, but the actual structure of these spectra is more varied, as illustrated by their Index functions. The tails of the electronic/positronic survival functions also largely deviate from simple power-law and lognormal distributions. The entropy of the electron/positron fluxes is calculated from the regressed broken power laws. Section 5 contains the conclusions.

In Appendix A, we summarize some elementary facts about power laws, lognormals and Weibull (stretched and compressed exponential) cutoffs in double-logarithmic representation, as well as their Index functionals representing the Log–Log slope of analytic distributions inferred by least-squares regression.

Multiply broken power-law distributions, their Index functions and cutoff factors

Broken power laws with smooth transitions

Broken power laws are defined as finite products [31, 32]

p(x)=A0xβ0k=1n(1+(x/bk)βk/|ηk|)ηk,
(2.1)

with positive amplitudes A0, bk, positive exponents βk, real exponent β0 and positive or negative exponents ηk. It is also understood that the factors are ordered by increasing amplitude bk<bk+1. The distribution p(x) consists of n+1 approximate power-law segments, xβ0,xβ0+β1sign(η1),…,xβ0+k=1nβksign(ηk), in the intervals x<<b1, b1<<x<<b2,,bn<<x, respectively. The amplitudes bk define the break points between the power-law segments and the exponents ηk determine the extent of the transitional regions; |ηk|<<1 implies a narrow transitional interval with a sudden change of slope at bk. Power laws in Log–Log plots show as approximately straight segments, and the exponents ηk determine the curvature of the transitions and the width of the transitional region between the power-law segments.

The Feller–Pareto (or generalized beta) distribution is a special case of (2.1), obtained by putting n=1 in the product, cf. Ref. [33]. This distribution consists of two power-law components, xβ0 for x<<b1 and xβ0+β1sign(η1) for x>>b1, and a smooth transition around the break point b1 and has been used to model income and wealth distributions [34, 35].

A practical way to quantify deviations from power-law scaling is provided by Index functions, occasionally employed in economic modeling, cf. e.g., Refs. [4, 5, 8, 34]. The Index function of a multiply broken power-law distribution (2.1) reads

Ip(x)=p(x)p(x)x=β0+k=1nsign(ηk)βk(x/bk)βk/|ηk|1+(x/bk)βk/|ηk|,
(2.2)

and gives the Log–Log slope of the distribution, cf. Appendix A for an elementary discussion of Index functions. In Sect. 3, we will employ Index functions to test power-law scaling predictions of the electrical resistivity of metals, arriving at the conclusion that the power-law scaling suggested by ab initio calculations based on various scattering mechanisms is not particularly accurate when compared with precision measurements of high-purity samples.

The introduction of multiply broken power-law distributions is motivated by the need to model ever bigger data sets stretching over several logarithmic decades, and standard textbook distributions [33] are insufficient to cope with that. A possible way to describe extended data sets is to use joint distributions, that is, to partition the data range into intervals and to use different standard distributions in each interval, joining them at the interval boundaries, cf. e.g., Refs. [36, 37]. Smoothness (differentiability) conditions at the interval boundaries then result in analytically rather complex distributions. The multiply broken power-law distributions (2.1) composed of structurally identical factors and the additive Index functions (2.2) offer a more uniform and technically simpler way to model extended data sets.

As pointed out above, the distribution (2.1) has power-law asymptotics, p(x)const.xβ0+k=1nβksign(ηk), for x>>bn. In the next section, we will discuss multiply broken power laws admitting exponential or lognormal decay, by adding a cutoff factor to the product (2.1), which will be used in Sect. 4 to model electronic and positronic particle fluxes.

Cutoff factors of multiply broken power laws

Weibull decay

A Weibull cutoff of the broken power law p(x) in (2.1) is obtained by adding the cutoff factor W(x)=exp((x/d)δ) to the product, with positive δ, d (and d>bn, see after (2.1)). The exponent δ defines stretched (0<δ<1) or compressed (δ>1) exponential decay. The Index function of W(x) is found as IW(x)=xW(x)/W(x)=δ(x/d)δ, to be added to the series in (2.2). That is, the Index function of the exponentially cut power law p(x)W(x) is IpW(x)=Ip(x)+IW(x).

Weibull cutoff with adaptable crossover

We generalize the above Weibull cutoff by defining, in analogy to the factors of the broken power law (2.1),

W(x)=exp[(1+(x/b)δ/η)η(d/b)δ+1(d/b)δ]exp[(x/d)δ+(b/d)δ],
(2.3)

which gives the standard Weibull cutoff in the limit b0, cf. Section 2.2.1. The second summand in the exponential is just a normalization factor, so that W(0)=1. The amplitudes b,d (d>bn, cf. (2.1)) and exponents δ, η are positive fitting parameters. W(x) is monotonously decaying, the crossover to the asymptotic regime depends on the exponent η, and the parameter b can be determined from the asymptotic amplitude factor exp((b/d)δ).

The Index function of W(x) in (2.3) reads

IW(x)=xW(x)W(x)=δ(x/b)δ/η1+(x/b)δ/η(1+(x/b)δ/η)η(d/b)δ,
(2.4)

and the Index function of the exponentially cut density p(x)W(x) is IpW(x)=Ip(x)+IW(x), where p(x) is the multiply broken power law (2.1) and Ip(x) and IW(x) are the Index functions in (2.2) and (2.4).

Lognormal decay

A lognormal cutoff of the multiply broken power law p(x) in (2.1) is generated by adding the factor

w(x)=(1+(x/b)β/|η|)ηδlog(1+(x/b)β/|η|)=(1+(x/b)β/|η|)ηexp[δlog2(1+(x/b)β/|η|)],
(2.5)

where b,β,δ are positive parameters and η is positive or negative. This cutoff is obtained by substituting (1+(x/b)β/|η|) for x in the lognormal xηδlogx, cf. Appendix A, according to the pattern (2.1). Asymptotically, w(x)A~xη~δ~logx, with constants A~, η~, δ~ depending on the fitting parameters b,β,δ,η. Also, w(0)=1. If the exponent η is negative, w(x) is monotonously decaying. If η is positive, w(x) attains a maximum before the decay sets in. The Index function of w(x) in (2.5) reads

Iw(x)=xw(x)w(x)=(x/b)β/|η|1+(x/b)β/|η|β|η|[η2δlog(1+(x/b)β/|η|)].
(2.6)

The Index function of the lognormally cut power law p(x)w(x), cf. (2.1) and (2.5), is obtained by adding the Index functions in (2.2) and (2.6), Ipw(x)=Ip(x)+Iw(x). Lognormal cutoffs have been used to model city-size and firm-size tail distributions, cf. e.g., Refs. [16,17,18,19] and references therein.

Analyticity of multiply broken power laws

The broken power-law density p(x) in (2.1) is analytic along the positive real axis but not at x=0, unless integer exponents are used, because of a branch cut along the negative real axis. In some applications, analyticity at x=0 is essential, for instance, when an ascending series expansion is required [38]. To make p(x) analytic at x=0, we replace xak+x in each factor, where the ak are positive fitting parameters, and also normalize each factor in (2.1) to one at x=0 except the first (with exponent β0). In this way, p(x) in (2.1) is transformed into

p(x)=A(a0+x)β0k=1n(1+((ak+x)/bk)βk/|ηk|)ηk(1+(ak/bk)βk/|ηk|)ηk.
(2.7)

The positive constants ak, like the amplitudes bk and the exponents βk,ηk, are to be determined by least-squares regression. The shift a0 can be put to zero from the outset if β0 is a positive integer, and the same holds true for ak if the exponent βk/|ηk| is integer, cf. after (2.1), but the exponents are usually real fitting parameters. The Index function Ip(x)=xp(x)/p(x) of density p(x) in (2.7) reads

Ip(x)=β0xa0+x+k=1nsign(ηk)βkxak+x((ak+x)/bk)βk/|ηk|1+((ak+x)/bk)βk/|ηk|,
(2.8)

obtained from (2.2) by shifting xak+x and adding a factor x/(ak+x) to each summand.

The Weibull cutoff factor defined in Sect. 2.2.1 can also be made analytic at x=0 (if δ is non-integer) by shifting xa+x, a>0, and normalizing, W(x)=exp(a/d)δexp[((a+x)/d)δ], so that W(0)=1. The Index function of W(x) is obtained from IW(x) in Sect. 2.2.1 by shifting xa+x and adding the factor x/(a+x), so that IW(x)=δ((a+x)/d)δx/(a+x).

The adaptable Weibull cutoff (2.3) can be made analytic in the same way, by replacing xa+x and subsequent normalization to W(0)=1,

W(x)=exp[(1+((a+x)/b)δ/η)η(d/b)δ+(1+(a/b)δ/η)η(d/b)δ].
(2.9)

The Index function of W(x) reads, cf. (2.4),

IW(x)=δxa+x((a+x)/b)δ/η1+((a+x)/b)δ/η(1+((a+x)/b)δ/η)η(d/b)δ.
(2.10)

Analogously, the lognormal cutoff w(x) in (2.5) becomes analytic at x=0 via the shift xa+x and normalization to w(0)=1,

w(x)=(1+((a+x)/b)β/|η|)ηδlog[1+((a+x)/b)β/|η|](1+(a/b)β/|η|)ηδlog(1+(a/b)β/|η|),
(2.11)

and admits the Index function

Iw(x)=xa+x((a+x)/b)β/|η|1+((a+x)/b)β/|η|β|η|{η2δlog[1+((a+x)/b)β/|η|]},
(2.12)

obtained from (2.6) by shifting xa+x and adding the factor x/(a+x), as in (2.10).

Modeling electrical resistivity data of metals by broken power-law distributions

The resistivity data used [22,23,24,25] cover several logarithmic decades in temperature, from the low-temperature regime up to the melting point. The classical Drude model of electrical resistivity predicts a linear temperature slope at high temperature. At low temperature, a T5 dependence is predicted by the Bloch–Grüneisen theory based on electron–phonon scattering, and a quadratic temperature dependence is suggested by electron–electron scattering, cf. the reviews [26, 27]. This refers to the ideal solid, discounting impurities. The actual low-temperature limit of the resistivity is constant (temperature independent) due to electron–impurity scattering.

We split the electrical resistivity into two components, ρ(T)=ρres+ρint(T), where ρres is the constant residual resistivity due to impurities. ρint(T) denotes the intrinsic temperature-dependent resistivity, modeled as a multiply broken power-law density (2.1). The parameters defining ρint(T) are obtained by least-square regression. In the examples below, we will study the temperature evolution of the electrical resistivity of copper, gold, chromium, aluminum, tungsten, iron and nickel.

The Index function Iρ(T) defining the Log–Log slope of the total resistivity ρ(T)=ρres+ρint(T) reads, cf. Section 2.1,

Iρ(T)=Tρ(T)ρ(T)=Iρint(T)ρint(T)ρres+ρint(T),Iρint(T):=Tρint(T)ρint(T),
(3.1)

where the Index Iρint(T) defines the slope of the intrinsic resistivity ρint(T). Thus, if the resistivity is dominated by electron–electron scattering in some low-temperature interval, this implies the Log–Log slope Iρint(T)2. If electron–phonon scattering is dominant at low temperature, Iρint(T)5. In the high-temperature regime, the Drude model predicts Iρint(T)Iρ(T)1.

Copper and gold

For the total electrical resistivity ρ(T)=ρres+ρint(T) of copper, we use the recommended data set of Ref. [22], which assumes 99.999% pure copper (annealed), and the data are corrected for thermal expansion. The least-squares regression in Fig. 1 is performed with a broken power law for the intrinsic resistivity, cf. (2.1) (with n=1),

ρint(T)=b0Tβ01(1+(T/b1)β1/η1)η1.
(3.2)
Fig. 1
figure1

Electrical resistivity of copper. Data points from Ref. [22]. The red solid curve depicts the total resistivity ρ(T)=ρres+ρint(T), where ρint(T) (blue dashed curve) is the intrinsic resistivity (modeled as broken power-law density (3.2)) and ρres the constant residual resistivity due to impurities. The fitting and goodness-of-fit parameters are listed in Table 1. ρint(T) is obtained by subtracting ρres (a fitting parameter) from the regressed total resistivity ρ(T). The temperature range extends up to the melting point at 1357.8 K. Residuals of the least-squares fit of ρ(T) are depicted in the lower panel

This is the simplest case of the broken power law (2.1), admitting only one break point b1, cf. Section 2.1, which is actually a standard distribution, a generalized beta distribution of the second kind, cf. Refs. [33,34,35]. The amplitudes and (positive) exponents are fitting parameters, as well as the residual resistivity ρres, cf. Table 1 where also the goodness-of-fit parameters are listed. The least-squares fit ρ(T) (red solid curve) and the intrinsic resistivity ρint(T) (dashed blue curve) of copper are depicted in Fig. 1. The Index function of the intrinsic resistivity ρint(T) in (3.2) reads, cf. (2.2),

Iρint(T)=β0β1(T/b1)β1/η11+(T/b1)β1/η1,
(3.3)
Table 1 Fitting and goodness-of-fit parameters of the electrical resistivity of copper, gold and chromium. The least-squares fits are performed with the total resistivity ρ(T)=ρres+ρint(T), cf. Section 3, and depicted in Figs. 1, 3 and 5. ρres is the constant residual resistivity due to impurity scattering of electrons, and ρint(T) is the intrinsic resistivity, modeled as a multiply broken power-law density. For copper and gold, ρint(T) is defined in (3.2), and for chromium in (3.4), with fitting parameters (amplitudes bi and exponents βi, ηi) recorded in this table. The residual resistivity ρres is also a free parameter in the least-squares fits. χ2 denotes the minimum of the least-squares functional χ2=i=1N(ρ(Ti)ρi)2/ρi2. The data points are recommended values from Refs. [22, 23]; no error bounds are given in these references, but the data are corrected for thermal expansion. The degrees of freedom (dof: number N of data points (Ti,ρi) minus number of independent fitting parameters), the standard error of the fit, SE=(i=1N(ρ(Ti)ρi)2/N)1/2, and the coefficient of determination R2=1i=1N(ρ(Ti)ρi)2/(Nσ2), with sample variance σ2=i=1N(ρiρ¯)2/N and mean ρ¯=i=1Nρi/N, are also recorded

which is depicted in Fig. 2 together with the Index Iρ(T) of the total resistivity, cf. (3.1).

Fig. 2
figure2

Index functions of copper. The red solid curve depicts the Index function Iρ(T) of the total resistivity in (3.1), calculated with the residual and intrinsic resistivities ρres and ρint(T) extracted from the least-squares fit in Fig. 1. The blue dashed curve shows the Index function Iρint(T) of the intrinsic resistivity, cf. (3.3). These Index functions represent the Log–Log slope of the resistivity curves in Fig. 1. At high temperature, the slope is nearly linear, with Index close to one (classical temperature scaling of resistivity). At low temperature, the Index of the intrinsic resistivity is close to five, indicating dominant electron–phonon scattering which predicts a T5 resistivity scaling

Figures 3 and 4 show the resistivities and the Index functions of gold (99.999% purity, annealed, data points [22] corrected for thermal expansion). The intrinsic resistivity ρint(T) and the least-squares fit of the total resistivity ρ(T)=ρres+ρint(T) are plotted in Fig. 3, based on the broken power law (3.2), like in the case of copper. The fitting parameters are stated in Table 1. The Index functions (3.1) and (3.3) of the total and intrinsic resistivities of gold are depicted in Fig. 4.

Fig. 3
figure3

Electrical resistivity of gold. Data points from Ref. [22]. The caption of Fig. 1 applies. The red solid curve depicts the least-squares fit of the total resistivity ρ(T)=ρres+ρint(T), cf. (3.2) and Table 1, with residual resistivity ρres and intrinsic resistivity ρint(T) (blue dashed curve). The fitting parameters are listed in Table 1. The temperature range is cut off at the melting point (1337.3 K). Residuals of the least-squares fit are depicted in the lower panel

Fig. 4
figure4

Index functions of gold. The caption of Fig. 2 applies. Depicted are the Index functions of the total electrical resistivity Iρ(T) (red solid curve) and intrinsic resistivity Iρint(T) (blue dashed curve) in (3.1) and (3.3), obtained from the least-squares fit in Fig. 3. At high temperature, the temperature scaling of the resistivity of gold is steeper than linear. The Index of the intrinsic resistivity is lower than four over the full temperature range of the solid phase; a T5 scaling of the intrinsic resistivity at low temperature predicted by the Bloch–Grüneisen theory based on electron–phonon scattering is not visible

As pointed out in Sect. 2.1 and Appendix A, the Index functions quantify the slope of the Log–Log resistivity curves. That is, a tangent to the resistivity curves represents a power law Tα, where the exponent α is the slope of the tangent line (in Log–Log coordinates) given by the Index function. In this section, we will use Index functions to check power-law scaling predictions of resistivity derived from electron–phonon scattering (Bloch–Grüneisen theory, suggesting a T5 power law for the intrinsic resistivity at low temperature) and electron–electron scattering (generating a low-temperature T2 contribution to the intrinsic resistivity). Which of these power laws is dominant also depends on the amplitudes. The Bloch–Grüneisen formula for the intrinsic resistivity of metals, which interpolates between the T5 scaling and the linear high-temperature scaling predicted by the classical Drude model of resistivity, is reviewed in Refs. [26, 27], including the derivation from linearized transport theory; for the latter, see also Refs. [39, 40] and references therein. Here, we will focus on the T2 and T5 scaling predictions by comparing with empirical data, and find that they are somewhat idealized.

The Index functions of gold and copper in Figs. 2 and 4 have a similar shape. At high temperature, the Index is only slightly above one (β0β1=1.12 for copper and β0β1=1.15 for gold, cf. (3.2), (3.3) and Table 1), so that the linear temperature scaling of the Drude model is approximately realized. In the low temperature regime, the Index Iρint(T) of the intrinsic resistivity of copper (blue dashed curve in Fig. 2) is noticeably above five (β0=5.34, cf. Table 1). For gold, the Index stays below four (β0=3.71), cf. Figure 4, so that a T5 low-temperature scaling of the intrinsic resistivity predicted by the Bloch–Grüneisen theory is not evidenced; a T2 slope suggested by electron–electron scattering is not visible either. Temperature intervals where simple power-law scaling Tα applies are indicated by nearly constant horizontal segments of the Index curves with ordinate α, cf. Appendix A. Power-law scaling in the high- and low-temperature regimes is clearly shown by the Index curves of the intrinsic resistivity in Figs. 2 and 4, albeit with exponents deviating from the scaling predictions of scattering theory.

Chromium

Figures 5 and 6 depict the least-squares fit of the resistivity and the Index function of chromium, based on recommended data points in Ref. [23] (99.98% purity, data corrected for thermal expansion). The χ2 fit of the total resistivity ρ(T)=ρres+ρint(T) is performed with the broken power-law density (2.1) (with