Stretched exponential function

Mathematical properties

Moments

Following the usual physical interpretation, we interpret the function argument t as time, and fβ(t) is the differential distribution. The area under the curve can thus be interpreted as a mean relaxation time. One finds \langle\tau\rangle \equiv \int_0^\infty dt, e^{-(t/\tau_K)^\beta} = {\tau_K \over \beta } \Gamma {\left( \frac 1 \beta \right)} where Γ is the gamma function. For exponential decay, is recovered.

The higher moments of the stretched exponential function are \langle\tau^n\rangle \equiv \int_0^\infty dt, t^{n-1}, e^{-(t/\tau_K)^\beta} = {{\tau_K}^n \over \beta }\Gamma {\left(\frac n \beta \right)}.

Distribution function

In physics, attempts have been made to explain stretched exponential behaviour as a linear superposition of simple exponential decays. This requires a nontrivial distribution of relaxation times, ρ(u), which is implicitly defined by e^{-t^\beta} = \int_0^\infty du,\rho(u), e^{-t/u}.

Alternatively, a distribution G = u \rho (u) is used.

ρ can be computed from the series expansion:{{cite journal | name-list-style=amp | year = 1980 For a more recent and general discussion, see {{cite journal \rho (u ) = -{ 1 \over \pi u} \sum_{k = 0}^\infty {(-1)^k \over k!} \sin (\pi \beta k)\Gamma (\beta k + 1) u^{\beta k}

For rational values of β, ρ(u) can be calculated in terms of elementary functions. But the expression is in general too complex to be useful except for the case where G(u) = u \rho(u) = { 1 \over 2\sqrt{\pi}} \sqrt{u} e^{-u/4}

Figure 2 shows the same results plotted in both a linear and a log representation. The curves converge to a Dirac delta function peaked at as β approaches 1, corresponding to the simple exponential function.

The moments of the original function can be expressed as \langle\tau^n\rangle = \Gamma(n) \int_0^\infty d\tau, t^n , \rho(\tau).

The first logarithmic moment of the distribution of simple-exponential relaxation times is \langle\ln\tau\rangle = \left( 1 - {1 \over \beta} \right) {\rm Eu} + \ln \tau_K where Eu is the Euler constant.{{cite journal

Fourier transform

To describe results from spectroscopy or inelastic scattering, the sine or cosine Fourier transform of the stretched exponential is needed. It must be calculated either by numeric integration, or from a series expansion. The series here as well as the one for the distribution function are special cases of the Fox–Wright function.{{cite journal | article-number = 061510 | doi-access = free

History and further applications

As said in the introduction, the stretched exponential was introduced by the German physicist Rudolf Kohlrausch in 1854 to describe the discharge of a capacitor (Leyden jar) that used glass as dielectric medium. The next documented usage is by Friedrich Kohlrausch, son of Rudolf, to describe torsional relaxation. A. Werner used it in 1907 to describe complex luminescence decays; Theodor Förster in 1949 as the fluorescence decay law of electronic energy donors.

Outside condensed matter physics, the stretched exponential has been used to describe the removal rates of small, stray bodies in the solar system,{{cite journal | display-authors=etal | doi-access = free}} and the production from unconventional gas wells.

In probability

If the integrated distribution is a stretched exponential, the normalized probability density function is given by p(\tau \mid \lambda, \beta)~d\tau = \frac{\lambda}{\Gamma(1 + \beta^{-1})} ~ e^{-(\tau \lambda)^\beta} ~ d\tau

Note that confusingly some authors have been known to use the name "stretched exponential" to refer to the Weibull distribution.{{cite book

Modified functions

A modified stretched exponential function f_\beta (t) = e^{ -t^{\beta(t)} } with a slowly t-dependent exponent β has been used for biological survival curves.{{cite journal |name-list-style=amp | year = 2009 | article-number = 19554

Wireless communications

In wireless communications, a scaled version of the stretched exponential function has been shown to appear in the Laplace Transform for the interference power I when the transmitters' locations are modeled as a 2D Poisson Point Process with no exclusion region around the receiver.{{cite book

The Laplace transform can be written for arbitrary fading distribution as follows: L_I(s) = \exp\left(-\pi \lambda \mathbb{E}{\left[g^\frac{2}{\eta} \right]} \Gamma{\left(1 - \frac{2}{\eta} \right)} s^\frac{2}{\eta}\right) = \exp\left(- t s^\beta \right) where g is the power of the fading, \eta is the path loss exponent, \lambda is the density of the 2D Poisson Point Process, \Gamma(\cdot) is the Gamma function, and \mathbb{E}[x] is the expectation of the variable x.

The same reference also shows how to obtain the inverse Laplace Transform for the stretched exponential \exp\left(-s^\beta \right) for higher order integer \beta = \beta_q \beta_b from lower order integers \beta_a and \beta_b.

Internet streaming

The stretched exponential has been used to characterize Internet media accessing patterns, such as YouTube and other stable streaming media sites. The commonly agreed power-law accessing patterns of Web workloads mainly reflect text-based content Web workloads, such as daily updated news sites.

References

References

1. (2021-06-14). "Slow stretched-exponential and fast compressed-exponential relaxation from local event dynamics". Journal of Physics: Condensed Matter.
2. Holm, Sverre. (2020). "Time domain characterization of the Cole-Cole dielectric model". Journal of Electrical Bioimpedance.
3. (2015). "Table of Integrals, Series, and Products". [[Academic Press, Inc.]].
4. Dishon et al. 1985.
5. (1 July 2002). "Friction on Small Objects and the Breakdown of Hydrodynamics in Solution: Rotation of Anthracene in Poly(isobutylene) from the Small-Molecule to Polymer Limits". The Journal of Physical Chemistry B.
6. (2010-01-01). "A Better Way To Forecast Production From Unconventional Gas Wells". Society of Petroleum Engineers.
7. Lei Guo, Enhua Tan, Songqing Chen, Zhen Xiao, and Xiaodong Zhang. (2008). ""The Stretched Exponential Distribution of Internet Media Access Patterns"".
8. (2000). "Power-Law Distribution of the World Wide Web". Science.