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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]].]]

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 exceptions of {{math|1=''β'' = 2}}, which gives the [[normal distribution]], and of compressed exponential relaxation in the dynamics of [[amorphous solids]].<ref>{{Cite journal |last1=Trachenko |first1=K. |last2=Zaccone |first2=A.|date=2021-06-14 |title=Slow stretched-exponential and fast compressed-exponential relaxation from local event dynamics |url=https://iopscience.iop.org/article/10.1088/1361-648X/ac04cd |journal=Journal of Physics: Condensed Matter |language=en |volume=33 |issue= |pages=315101 |doi= 10.1088/1361-648X/ac04cd|bibcode= |issn=0953-8984|arxiv=2010.10440 }}</ref>

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]].

In physics, the stretched exponential function is often used as a phenomenological description of [[Relaxation (physics)|relaxation]] in disordered systems. It was first introduced by [[Rudolf Kohlrausch]] in 1854 to describe the discharge of a capacitor;<ref>{{cite journal
| author = Kohlrausch, R.
| year = 1854
| title = Theorie des elektrischen Rückstandes in der Leidner Flasche
| journal = [[Annalen der Physik und Chemie]]
| volume = 91
| issue = 1
| pages = 56–82, 179–213
| url = http://gallica.bnf.fr/ark:/12148/bpt6k15176w.pagination| doi = 10.1002/andp.18541670103
| bibcode = 1854AnP...167...56K
}}.</ref> 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 spectroscopy|dielectric spectra]] of polymers;<ref>{{cite journal
|author1=Williams, G. |author2=Watts, D. C.
|name-list-style=amp | year = 1970
| title = Non-Symmetrical Dielectric Relaxation Behavior Arising from a Simple Empirical Decay Function
| journal = Transactions of the Faraday Society
| volume = 66
| pages = 80–85
| 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 [[Cole–Cole equation]], the [[Cole–Davidson equation]], and the [[Havriliak–Negami relaxation]], for small time arguments.<ref>{{Cite journal |last=Holm|first=Sverre|title=Time domain characterization of the Cole-Cole dielectric model |journal=Journal of Electrical Bioimpedance |year=2020 |volume=11 |issue=1 |pages=101–105|doi=10.2478/joeb-2020-0015|pmid=33584910 |pmc=7851980 }}</ref>

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.
| volume = 28
| pages = 1–47
| year = 1975
| title = Asymptotic evaluation of certain Markov process expectations for large time
| doi=10.1002/cpa.3160280102
}}</ref><ref>{{cite journal
| author = Takano, H. and Nakanishi, H. and Miyashita, S.
| journal = Phys. Rev. B
| volume = 37
| issue = 7
| pages = 3716–3719
| year = 1988
| title = Stretched exponential decay of the spin-correlation function in the kinetic Ising model below the critical temperature
|bibcode = 1988PhRvB..37.3716T |doi = 10.1103/PhysRevB.37.3716 | pmid = 9944981
}}</ref><ref>{{cite journal
| author = Shore, John E. and Zwanzig, Robert
| journal = The Journal of Chemical Physics
| volume = 63
| issue = 12
| pages = 5445–5458
| year = 1975
| title = Dielectric relaxation and dynamic susceptibility of a one-dimensional model for perpendicular-dipole polymers
|doi = 10.1063/1.431279| bibcode = 1975JChPh..63.5445S
}}</ref><ref>{{cite journal
| author = Brey, J. J. and Prados, A.
| journal = Physica A
| volume = 197
| issue = 4
| pages = 569–582
| year = 1993
| title = Stretched exponential decay at intermediate times in the one-dimensional Ising model at low temperatures
|doi = 10.1016/0378-4371(93)90015-V | bibcode = 1993PhyA..197..569B
}}</ref> it can be shown that the asymptotic decay is a stretched exponential, but the prefactor is usually an unrelated power.

== Mathematical properties ==

=== 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 |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
<math display="block">e^{-t^\beta} = \int_0^\infty du\,\rho(u)\, e^{-t/u}.</math>

Alternatively, a distribution <math display="block">G = u \rho (u)</math> is used.

''ρ'' can be computed from the series expansion:<ref>{{cite journal
| author1=Lindsey, C. P. | author2=Patterson, G. D.
| name-list-style=amp | year = 1980
| title = Detailed comparison of the Williams-Watts and Cole-Davidson functions
| journal = [[Journal of Chemical Physics]]
| volume = 73
| issue = 7
| pages = 3348–3357
| doi = 10.1063/1.440530 | bibcode = 1980JChPh..73.3348L }}.
For a more recent and general discussion, see {{cite journal
| author = Berberan-Santos, M.N., Bodunov, E.N. and Valeur, B.
| year = 2005
| title = Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential)
| journal = [[Chemical Physics]]
| volume = 315
| issue = 1–2
| pages = 171–182
| doi = 10.1016/j.chemphys.2005.04.006
|bibcode = 2005CP....315..171B }}.</ref>
<math display="block"> \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}</math>

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 {{math|1=''β'' = 1/2}} where
<math display="block">G(u) = u \rho(u) = { 1 \over 2\sqrt{\pi}} \sqrt{u} e^{-u/4}
</math>

Figure 2 shows the same results plotted in both a [[linear]] and a [[Logarithm|log]] representation. The curves converge to a [[Dirac delta function]] peaked at {{math|1=''u'' = 1}} as ''β'' approaches 1, corresponding to the simple exponential function.
{| class="wikitable" style="margin: 1em auto 1em auto"
|-
| [[Image:KWW dist. function linear.png|300px]] || [[Image:KWW dist. funct. log.png|300px]]
|-
| colspan=2 | '''Figure 2'''. Linear and log-log plots of the stretched exponential distribution function <math>G</math> vs <math>t/\tau</math>
for values of the stretching parameter ''β'' between 0.1 and 0.9.
|}

The moments of the original function can be expressed as
<math display="block">\langle\tau^n\rangle = \Gamma(n) \int_0^\infty d\tau\, t^n \, \rho(\tau).</math>

The first logarithmic moment of the distribution of simple-exponential relaxation times is
<math display="block">\langle\ln\tau\rangle = \left( 1 - {1 \over \beta} \right) {\rm Eu} + \ln \tau_K </math>
where Eu is the [[Euler constant]].<ref>{{cite journal
| doi = 10.1063/1.1446035
| author = Zorn, R.
| year = 2002
| title = Logarithmic moments of relaxation time distributions
| journal = [[Journal of Chemical Physics]]
| volume = 116
| issue = 8
| pages = 3204–3209
|bibcode = 2002JChPh.116.3204Z | url = http://juser.fz-juelich.de/record/1954/files/10418.pdf
}}</ref>

== 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.<ref>Dishon et al. 1985.</ref> The series here as well as the one for the distribution function are special cases of the [[Fox–Wright function]].<ref>{{cite journal
| author = Hilfer, J.
| year = 2002
| title = ''H''-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems
| journal = Physical Review E
| volume = 65
| issue = 6
| pages = 061510
| doi = 10.1103/physreve.65.061510
| pmid = 12188735
| bibcode = 2002PhRvE..65f1510H
| s2cid = 16276298
}}</ref> For practical purposes, the Fourier transform may be approximated by the [[Havriliak–Negami relaxation|Havriliak–Negami function]],<ref>{{cite journal
| author = Alvarez, F., Alegría, A. and Colmenero, J.
| year = 1991
| title = Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions
| journal = Physical Review B
| volume = 44
| issue = 14
| pages = 7306–7312
| doi = 10.1103/PhysRevB.44.7306
| pmid = 9998642
| bibcode = 1991PhRvB..44.7306A }}</ref> though nowadays the numeric computation can be done so efficiently<ref>{{cite journal
| author = Wuttke, J.
| year = 2012
| title = Laplace–Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source Implementation "libkww"
|journal = [[Algorithms (journal)|Algorithms]]
| volume = 5
| issue = 4
| pages = 604–628
| doi = 10.3390/a5040604 | arxiv = 0911.4796
| s2cid = 15030084
| doi-access = free
}}</ref> 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 [[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
| title = Lifetimes of small bodies in planetocentric (or heliocentric) orbits
| journal = [[Icarus (journal)|Icarus]]
| volume = 188
| issue = 2
| pages = 481–505
| doi = 10.1016/j.icarus.2006.11.024 | bibcode = 2007Icar..188..481D }}</ref> the diffusion-weighted MRI signal in the brain,<ref>{{cite journal
| author = Bennett, K. | year = 2003
| title = Characterization of Continuously Distributed Water Diffusion Rates in Cerebral Cortex with a Stretched Exponential Model
| journal = Magn. Reson. Med.
| volume = 50
| issue = 4
| pages = 727–734
| doi = 10.1002/mrm.10581 | pmid = 14523958
| 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 [[probability distribution|probability density function]] is given by{{Citation needed|date=May 2023}}
<math display="block"> p(\tau \mid \lambda, \beta)~d\tau = \frac{\lambda}{\Gamma(1 + \beta^{-1})} ~ e^{-(\tau \lambda)^\beta} ~ d\tau</math>

Note that confusingly some authors have been known to use the name "stretched exponential" to refer to the [[Weibull distribution]].<ref>{{cite book
| author = Sornette, D.
| year = 2004
| title = Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder}}.</ref>

=== Modified functions ===

A modified stretched exponential function
<math display="block">f_\beta (t) = e^{ -t^{\beta(t)} }</math>
with a slowly ''t''-dependent exponent ''β'' has been used for biological survival curves.<ref>{{cite journal
|author1=B. M. Weon |author2=J. H. Je
|name-list-style=amp | year = 2009
| title = Theoretical estimation of maximum human lifespan
| journal = Biogerontology
| volume = 10
| issue = 1
| pages = 65–71
| doi = 10.1007/s10522-008-9156-4 | pmid=18560989
|s2cid=8554128
}}</ref><ref>
{{cite journal
| author = B. M. Weon
| year = 2016
| title = Tyrannosaurs as long-lived species
| journal = Scientific Reports
| volume = 6
| pages = 19554
| doi = 10.1038/srep19554
| pmid=26790747
| pmc=4726238| bibcode = 2016NatSR...619554W
}}</ref>

=== 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 <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.
| title = 2018 IEEE International Conference on Communications (ICC)
| chapter = Closed Form Expressions for the Probability Density Function of the Interference Power in PPP Networks
| year = 2018
| pages = 1–6
| doi = 10.1109/ICC.2018.8422214 | arxiv = 1803.10440 | isbn = 978-1-5386-3180-5
| s2cid = 4374550
}}</ref>

The [[Laplace transform]] can be written for arbitrary [[fading]] distribution as follows:
<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}}

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 Streaming ==

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 ==
<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

{{DEFAULTSORT:Stretched Exponential Function}}
[[Category:Exponentials]]

Stretched exponential function - Revision history

imported>Ellie WombatWhisperer: /* growthexperiments-addlink-summary-summary:3|0|0 */