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{{Short description|Mathematical function common in physics}}
[[Image:Pibmasterplot.png|325px|thumb|'''Figure 1'''. Illustration of a stretched exponential fit (with ''β''=0.52) to an empirical master curve. For comparison, a least squares single and a [[Double exponential function|double exponential]] fit are also shown. The data are rotational [[anisotropy]] of [[anthracene]] in [[polyisobutylene]] of several [[molecular mass]]es. The plots have been made to overlap by dividing time (''t'') by the respective characteristic [[time constant]].]]▼
[[File:Stretched exponential.svg|325px|thumb|'''Figure 1'''. Plot of {{math|1=''f''<sub>''β''</sub>(''t'') = ''e''<sup>−''t''<sup>''β''</sup></sup>}} for varying values of ''β'', with stretched exponentials ({{math|1=''β'' < 1}}) in reddish colors, compressed exponentials ({{math|1=''β'' > 1}}) in green and blue colors, and the standard exponential function in yellow. The [[degenerate case]]s {{math|1=''β'' → 0}} and {{math|1=''β'' → +∞}} are marked in dotted lines.]]
The '''stretched exponential function''' <math display="block">f_\beta (t) = e^{ -t^\beta }</math> is obtained by inserting a fractional [[power law]] into the [[exponential function]]. In most applications, it is meaningful only for arguments {{mvar|t}} between 0 and +∞. With {{math|1=''β'' = 1}}, the usual exponential function is recovered. With a ''stretching exponent'' ''β'' 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 {{math|1=''β'' > 1}}) has less practical importance, with the notable
▲In most applications, it is meaningful only for arguments {{mvar|t}} between 0 and +∞. With {{math|1=''β'' = 1}}, the usual exponential function is recovered. With a ''stretching exponent'' ''β'' 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 {{math|1=''β'' > 1}}) has less practical importance, with the notable exception of {{math|1=''β'' = 2}}, which gives the [[normal distribution]].
In mathematics, the stretched exponential is also known as the [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|complementary cumulative]] [[Weibull distribution]]. The stretched exponential is also the [[characteristic function (probability theory)|characteristic function]], basically the [[Fourier transform]], of the [[stable distribution|Lévy symmetric alpha-stable distribution]].
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| doi = 10.1039/tf9706600080
|s2cid=95007734
}}.</ref> in this context, the stretched exponential or its Fourier transform are also called the '''Kohlrausch–Williams–Watts (KWW) function'''. The Kohlrausch–Williams–Watts (KWW) function corresponds to the time ___domain charge response of the main dielectric models, such as the [[
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,<ref>{{cite journal
|author1=Donsker, M. D. |author2=Varadhan, S. R. S.
|name-list-style=amp | journal = Comm. Pure Appl. Math.
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=== Moments ===
Following the usual physical interpretation, we interpret the function argument ''t'' as time, and ''f''<sub>β</sub>(''t'') is the differential distribution. The area under the curve can thus be interpreted as a ''mean relaxation time''. One finds
<math display="block">\langle\tau\rangle \equiv \int_0^\infty dt\, e^{-(t/\tau_K)^\beta} = {\tau_K \over \beta } \Gamma {\left( \frac 1 \beta \right)}</math>
where {{math|Γ}} is the [[gamma function]]. For [[exponential decay]], {{math|1=⟨''τ''⟩ = ''τ''<sub>''K''</sub>}} is recovered.
The higher [[moment (mathematics)|moments]] of the stretched exponential function are<ref name="Zwillinger_2014">{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=English |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik |chapter=3.478. |page=372}}</ref>
<math display="block">\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)}.</math>
=== 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
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== History and further applications ==
▲[[Image:Pibmasterplot.png|325px|thumb|'''Figure
As said in the introduction, the stretched exponential was introduced by the [[Germans|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 (physicist)|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.{{Citation needed|date=May 2023}}
Outside [[condensed matter physics]], the stretched exponential has been used to describe the removal rates of small, stray bodies in the solar system,<ref>{{cite journal
| author = Dobrovolskis, A., Alvarellos, J. and Lissauer, J.
| year = 2007
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| display-authors=etal | doi-access = free}}</ref> and the production from unconventional gas wells.<ref>{{Cite journal |last1=Valko|first1=Peter P. |last2=Lee|first2=W. John |date=2010-01-01 |title=A Better Way To Forecast Production From Unconventional Gas Wells|journal=SPE Annual Technical Conference and Exhibition |language=english | publisher=Society of Petroleum Engineers | doi=10.2118/134231-ms | isbn=9781555633004}}</ref>
=== In probability
If the integrated distribution is a stretched exponential, the normalized [[
<math display="block"> p(\tau \mid \lambda, \beta)~d\tau = \frac{\lambda}{\Gamma(1 + \beta^{-1})} ~ e^{-(\tau \lambda)^\beta} ~ d\tau</math>
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}}</ref>
=== Wireless
In wireless communications, a scaled version of the stretched exponential function has been shown to appear in the Laplace Transform for the interference power <math>I</math> when the transmitters' locations are modeled as a 2D [[Poisson point process|Poisson Point Process]] with no exclusion region around the receiver.<ref>{{cite book
| author = Ammar, H. A., Nasser, Y. and Artail, H.
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}}</ref>
The [[Laplace
<math display="block"> 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)</math>
where <math>g</math> is the power of the fading, <math>\eta</math> is the [[Path loss#Loss exponent|path loss exponent]], <math>\lambda</math> is the density of the 2D Poisson Point Process, <math>\Gamma(\cdot)</math> is the Gamma function, and <math>\mathbb{E}[x]</math> is the expectation of the variable <math>x</math>.{{Citation needed|date=May 2023}}
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The same reference also shows how to obtain the inverse Laplace Transform for the stretched exponential <math>\exp\left(-s^\beta \right)</math> for higher order integer <math>\beta = \beta_q \beta_b </math> from lower order integers <math>\beta_a</math> and <math>\beta_b</math>.{{Citation needed|date=May 2023}}
=== Internet
The stretched exponential has been used to characterize Internet media accessing patterns, such as YouTube and other stable [[streaming media]] sites.<ref>{{Cite conference |author= Lei Guo, Enhua Tan, Songqing Chen, Zhen Xiao, and Xiaodong Zhang| title="The Stretched Exponential Distribution of Internet Media Access Patterns" |conference= PODC' 08| pages=283–294|year=2008 | doi=10.1145/1400751.1400789 }}</ref> The commonly agreed power-law accessing patterns of Web workloads mainly reflect text-based content Web workloads, such as daily updated news sites.<ref>{{
==References==▼
▲== References ==
<references/>
== External links ==
* J. Wuttke: [http://apps.jcns.fz-juelich.de/kww libkww] C library to compute the Fourier transform of the stretched exponential function
[[Category:Exponentials]]
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