Stretched exponential function: Difference between revisions

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{{Short description|Mathematical function common in physics}}

[[File:Stretched exponential.svg|325px|thumb|'''Figure 1'''. Plot of {{math|1=''f''_''β''(''t'') = ''e''^{−''t''^''β''}}} 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.]]

[[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&nbsp;''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>

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== History and further applications ==

[[Image:Pibmasterplot.png|325px|thumb|'''Figure 13'''. 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.<ref>{{cite journal |last1=Sluch |first1=Mikhail I. |last2=Somoza |first2=Mark M. |last3=Berg |first3=Mark A. |title=Friction on Small Objects and the Breakdown of Hydrodynamics in Solution: Rotation of Anthracene in Poly(isobutylene) from the Small-Molecule to Polymer Limits |journal=The Journal of Physical Chemistry B |date=1 July 2002 |volume=106 |issue=29 |pages=7385–7397 |doi=10.1021/jp025549u}}</ref> The plots have been made to overlap by dividing time (''t'') by the respective characteristic [[time constant]].]]

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}}

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=== Wireless Communicationscommunications ===

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 Streamingstreaming ===

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>

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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

{{DEFAULTSORT:Stretched Exponential Function}}

[[Category:Exponentials]]