Stretched exponential function: Difference between revisions

Revision as of 04:51, 16 December 2019

Revision as of 04:53, 16 December 2019

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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 applybe used to describe the differential or to the integral distribution function—or to 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.

|lastauthoramp=yes | journal = Comm. Pure Appl. Math.

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=== Moments ===

Following the usual physical interpretation, we interpret the function argument ''t'' as a time, and ''f''_β(''t'') is the differential distribution. The area under the curve

iscan thereforethus be interpreted as a ''mean relaxation time''. One finds

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

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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=http://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik |chapter=3.478. |page=372}}</ref>