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

Appendix A: Log–Log and Lin-Log coordinates, power-law slopes, lognormals, and Index functions

A positive function $y = f(x)$ in linear coordinates $x,y \ge 0$ is transformed into logarithmic coordinates (denoted by a hat) $x = 10^{{\hat{x}}}$, $y = 10^{{\hat{y}}}$ by

$$ \hat{y} = {\text{Log}}(f(10^{{\hat{x}}} )) = \frac{{\log (f({\text{e}}^{{\hat{x}\log 10}} ))}}{\log 10}, $$

(A.1)

where ${\text{Log(}}x) = \log x/\log 10$ denotes the decadic logarithm. Conversely, a function $\hat{y} = g(\hat{x})$ in Log–Log representation is converted via $\hat{x} = {\text{Log(}}x)$, $\hat{y} = {\text{Log(}}y)$ into linear coordinates $x,y \ge 0$ by

$$ y = 10^{{g({\text{Log}}(x))}} = \exp \left[ {g(\frac{\log x}{{\log 10}})\log 10} \right]. $$

(A.2)

To obtain a tangent line in a double-logarithmic plot, we note $\hat{y}^{\prime}(\hat{x}) = xf^{\prime}(x)/f(x)$, according to (A.1). The Log–Log tangent at $\hat{x}_{0} = \log x_{0} /\log 10$, $\hat{y}_{0} = \log f(x_{0} )/\log 10$, reads $\hat{y} = \alpha \hat{x} + a$, with slope $\alpha = x_{0} f^{\prime}(x_{0} )/f(x_{0} )$ and vertical intercept $a = (\log f(x_{0} ) - \alpha \log x_{0} )/\log 10$. This Log–Log tangent corresponds to the power law $y = 10^{a} x^{\alpha }$ in linear coordinates, cf. (A.2). As for Log–Log inflection points, we note $\hat{y}^{\prime\prime}(\hat{x})/\log 10 = (xf^{\prime}(x)/f(x))^{\prime}x$; if $\hat{x}_{0}$ is an inflection point of a Log–Log plot, the derivative of $xf^{\prime}(x)/f(x)$ vanishes at $x_{0} = 10^{{\hat{x}_{0} }}$. Local minima and maxima of $xf^{\prime}(x)/f(x)$ thus indicate inflection points of $\hat{y}(\hat{x})$ in (A.1). In general, a Log–Log inflection point does not have a counterpart in linear representation.

Example 1

$y = Ax^{{\alpha + \beta \log x + \gamma \log^{2} x + \cdots }}$, where the ellipsis in the exponent indicates possible higher powers of $\log x$, so that $y^{\prime} = (\alpha + 2\beta \log x + 3\gamma \log^{2} x + \cdots )y/x$. If $\gamma = 0$, $\beta < 0$, this is a lognormal distribution. In Log–Log coordinates, we obtain a polynomial,

$$ \hat{y} = \frac{\log A}{{\log 10}} + \alpha \hat{x} + \beta \hat{x}^{2} \log 10 + \gamma \hat{x}^{3} \log^{2} 10 + \cdots . $$

(A.3)

Lognormals are second-order polynomials in Log–Log representation, and a power law ($\beta = \gamma = 0$) is Log–Log linear. Thus, the regression of a lognormal distribution just amounts to fitting a second-order polynomial to a Log–Log plot of the data set, and a power law implies linear regression.

Example 2

$y = Ax^{\alpha } \exp ( - (x/d)^{\delta } )$, with $A,d,\delta > 0$, so that $y^{\prime} = (\alpha - \delta (x/d)^{\delta } )y/x$. In Log–Log coordinates, this Weibull distribution reads

$$ \hat{y} = \frac{\log A}{{\log 10}} + \alpha \hat{x} - \frac{{{\text{e}}^{{\delta \hat{x}\log 10}} }}{{d^{\delta } \log 10}}, $$

(A.4)

the power-law factor $Ax^{\alpha }$ becomes a linear function, and the cutoff factor emerges as subtracted exponential in this representation.

To define Index functionals, we start with a function $y = f(x)$, $x > 0$, which reads, in Lin-Log coordinates, $y = f(10^{{\hat{x}}} )$, the logarithmic coordinate being denoted by a hat, $\hat{x} = {\text{Log(}}x)$. The function $y = I_{f} (x) = xf^{\prime}(x)/f(x)$ is the linear representation of the Log–Log slope (Index) of $f(x)$, see after (A.2). The Lin-Log representation of the Index $I_{f} (x)$ is $y = I_{f} (10^{{\hat{x}}} ) = 10^{{\hat{x}}} f^{\prime}(10^{{\hat{x}}} )/f(10^{{\hat{x}}} )$.

For instance, a lognormal $f(x) = Ax^{\alpha + \beta \log x}$, $\beta < 0$, has the Index function $y = I_{f} (10^{{\hat{x}}} ) = \alpha + 2\beta \hat{x}\log 10$. Thus, the Index of a lognormal is linear in Lin-Log representation, and the Index of a power law is constant. Conversely, if the Index $I_{f} (x)$ is constant or linear (with negative slope) in Lin-Log representation $y = I_{f} (10^{{\hat{x}}} )$, then $f(x)$ is a power law or lognormal, cf. Example 1 above, where $y = I_{f} (x) = \alpha + 2\beta \log x + \cdots$. A Weibull distribution, $f(x) = Ax^{\alpha } \exp ( - (x/d)^{\delta } )$, has the Index function $y = I_{f} (x) = \alpha - \delta (x/d)^{\delta } = \alpha - \delta {\text{e}}^{{\delta \hat{x}\log 10}} /d^{\delta }$, cf. Example 2 above.