Stretched exponential function: Difference between revisions

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Revision as of 06:41, 19 October 2020

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| 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

}}.</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

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| pages = 80–85

| doi = 10.1039/tf9706600080

}}.</ref> in this context, the stretched exponential or its Fourier transform are also called the '''Kohlrausch–Williams–Watts (KWW) function'''.

}}.</ref>

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 to neither.

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| 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

<ref>{{cite journal

| author = Shore, John E. and Zwanzig, Robert

| journal = The Journal of Chemical Physics

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| 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

<ref>{{cite journal

| author = Brey, J. J. and Prados, A.

| journal = Physica A

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| title = Stretched exponential decay at intermediate times in the one-dimentional 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.

}}</ref>

it can be shown that the asymptotic decay is a stretched exponential, but the prefactor is usually an unrelated power.

== Mathematical properties ==

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where Γ is the [[gamma function]]. For exponential decay, 〈''τ''〉&nbsp;=&nbsp;''τ''_''K'' 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 |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>

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 |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>\langle\tau^n\rangle \equiv \int_0^\infty dt\, t^{n-1}\, e^{-(t/\tau_K)^\beta} = {{\tau_K}^n \over \beta }\Gamma \left({n \over \beta }\right).</math>

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| 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

though nowadays the numeric computation can be done so efficiently<ref>{{cite journal

| author = Wuttke, J.

| year = 2012

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| author = Sornette, D.

| year = 2004

| title = Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder}}.</ref> have been known to use the name "stretched exponential" to refer to the [[Weibull distribution]].

have been known to use the name "stretched exponential" to refer to the [[Weibull distribution]].

=== Modified functions ===

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| doi = 10.1007/s10522-008-9156-4|pmid=18560989

|s2cid=8554128

}}</ref><ref>

<ref>

{{cite journal

| author = B. M. Weon