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is obtained by inserting a fractional power law into the exponential function.
In most applications, it is meaningful only for arguments t between 0 and +?. With ? = 1, the usual exponential function is recovered. With a stretching exponent? between 0 and 1, the graph of log f versus t is characteristically stretched, hence the name of the function. The compressed exponential function (with ? > 1) has less practical importance, with the notable exception of ? = 2, which gives the normal distribution.
In physics, the stretched exponential function is often used as a phenomenological description of relaxation in disordered systems. It was first introduced by Rudolf Kohlrausch in 1854 to describe the discharge of a capacitor; thus it is also known as the Kohlrausch function. In 1970, G. Williams and D.C. Watts used the Fourier transform of the stretched exponential to describe dielectric spectra of polymers; in this context, the stretched exponential or its Fourier transform are also called the Kohlrausch-Williams-Watts (KWW) function.
In phenomenological applications, it is often not clear whether the stretched exponential function should be used to describe the differential or the integral distribution function--or neither.
In each case, one gets the same asymptotic decay, but a different power law prefactor, which makes fits more ambiguous than for simple exponentials. In a few cases, it can be shown that the asymptotic decay is a stretched exponential, but the prefactor is usually an unrelated power.
Following the usual physical interpretation, we interpret the function argument t as time, and f?(t) is the differential distribution. The area under the curve
can thus be interpreted as a mean relaxation time. One finds
where ? is the gamma function. For exponential decay, ??? = ?K is recovered.
The higher moments of the stretched exponential function are
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, ?(u), which is implicitly defined by
For rational values of ?, ?(u) can be calculated in terms of elementary functions. But the expression is in general too complex to be useful except for the case ? = 1/2 where
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 u = 1 as ? approaches 1, corresponding to the simple exponential function.
Figure 2. Linear and log-log plots of the stretched exponential distribution function vs
for values of the stretching parameter ? between 0.1 and 0.9.
The moments of the original function can be expressed as
The first logarithmic moment of the distribution of simple-exponential relaxation times is
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. For practical purposes, the Fourier transform may be approximated by the Havriliak-Negami function, though nowadays the numeric computation can be done so efficiently that there is no longer any reason not to use the Kohlrausch-Williams-Watts function in the frequency domain.
History and further applications
As said in the introduction, the stretched exponential was introduced by the GermanphysicistRudolf 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, the diffusion-weighted MRI signal in the brain, and the production from unconventional gas wells.
Note that confusingly some authors have been known to use the name "stretched exponential" to refer to the Weibull distribution.
A modified stretched exponential function
with a slowly t-dependent exponent β has been used for biological survival curves.
In wireless communications, a scaled version of the stretched exponential function has been shown to appear in the Laplace Transform for the interference power when the transmitters' locations are modeled as a 2D Poisson Point Process with no exclusion region around the receiver.
^Williams, G. & Watts, D. C. (1970). "Non-Symmetrical Dielectric Relaxation Behavior Arising from a Simple Empirical Decay Function". Transactions of the Faraday Society. 66: 80-85. doi:10.1039/tf9706600080..
^Donsker, M. D. & Varadhan, S. R. S. (1975). "Asymptotic evaluation of certain Markov process expectations for large time". Comm. Pure Appl. Math. 28: 1-47. doi:10.1002/cpa.3160280102.
^Shore, John E. and Zwanzig, Robert (1975). "Dielectric relaxation and dynamic susceptibility of a one-dimensional model for perpendicular-dipole polymers". The Journal of Chemical Physics. 63 (12): 5445-5458. Bibcode:1975JChPh..63.5445S. doi:10.1063/1.431279.CS1 maint: multiple names: authors list (link)
^Valko, Peter P.; Lee, W. John (2010-01-01). "A Better Way To Forecast Production From Unconventional Gas Wells". SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. doi:10.2118/134231-ms. ISBN9781555633004.
^Sornette, D. (2004). Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder..
^Ammar, H. A., Nasser, Y. and Artail, H. (2018). "Closed Form Expressions for the Probability Density Function of the Interference Power in PPP Networks". 2018 IEEE International Conference on Communications (ICC): 1-6. arXiv:1803.10440. doi:10.1109/ICC.2018.8422214.CS1 maint: multiple names: authors list (link)
J. Wuttke: libkww C library to compute the Fourier transform of the stretched exponential function