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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 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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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>
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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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}}</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
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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>{{cite journal |last1=Adamic|first1=Lada A. |last2=Bernardo A. |first2=Huberman |year=2000 |title=Power-Law Distribution of the World Wide Web |url= |journal=Science |volume=287 |issue=5461 |pages=2115-2115 |doi=10.1126/science.287.5461.2115a}}</ref>
== References ==
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* 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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