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 display|1="block"''f''_''β''f_\beta (''t'') = ''e^{ -''<sup>−''t^\beta }''^{''β''</mathsup>}}} for varying values of ''β'', with stretched exponentials ({{math|1=''β'' < 1}}) in warmreddish colors, compressed exponentials ({{math|1=''β'' > 1}}) in coolgreen and blue colors, and the standard exponential function in yellow. The limiting[[degenerate casescase]]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&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>