Stretched exponential function

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Mathematical function common in physics

title: "Stretched exponential function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["exponentials"] description: "Mathematical function common in physics" topic_path: "general/exponentials" source: "https://en.wikipedia.org/wiki/Stretched_exponential_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical function common in physics ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/cb/Stretched_exponential.svg" caption="1=''β'' → +∞}} are marked in dotted lines."] ::

The stretched exponential function f_\beta (t) = e^{ -t^\beta } is obtained by inserting a fractional power law into the exponential function. In most applications, it is meaningful only for arguments t between 0 and +∞. With , 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 ) has less practical importance, with the notable exceptions of , which gives the normal distribution, and of compressed exponential relaxation in the dynamics of amorphous solids.

In mathematics, the stretched exponential is also known as the complementary cumulative Weibull distribution. The stretched exponential is also the characteristic function, basically the Fourier transform, of the Lévy symmetric alpha-stable distribution.

In physics, the stretched exponential function is often used as a phenomenological description of relaxation in disordered systems. It was first introduced by Rudolf Kohlrausch in 1854 to describe the discharge of a capacitor;{{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 |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

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,{{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 | 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 | 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 | 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

Mathematical properties

Moments

Following the usual physical interpretation, we interpret the function argument t as time, and fβ(t) is the differential distribution. The area under the curve can thus be interpreted as a mean relaxation time. One finds \langle\tau\rangle \equiv \int_0^\infty dt, e^{-(t/\tau_K)^\beta} = {\tau_K \over \beta } \Gamma {\left( \frac 1 \beta \right)} where Γ is the gamma function. For exponential decay, is recovered.

The higher moments of the stretched exponential function are \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)}.

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 e^{-t^\beta} = \int_0^\infty du,\rho(u), e^{-t/u}.

Alternatively, a distribution G = u \rho (u) is used.

ρ can be computed from the series expansion:{{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 }}. \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}

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 where G(u) = u \rho(u) = { 1 \over 2\sqrt{\pi}} \sqrt{u} e^{-u/4}

Figure 2 shows the same results plotted in both a linear and a log representation. The curves converge to a Dirac delta function peaked at as β approaches 1, corresponding to the simple exponential function. ::data[format=table]

Figure 2. Linear and log-log plots of the stretched exponential distribution function G vs t/\tau
::

The moments of the original function can be expressed as \langle\tau^n\rangle = \Gamma(n) \int_0^\infty d\tau, t^n , \rho(\tau).

The first logarithmic moment of the distribution of simple-exponential relaxation times is \langle\ln\tau\rangle = \left( 1 - {1 \over \beta} \right) {\rm Eu} + \ln \tau_K where Eu is the Euler constant.{{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

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. The series here as well as the one for the distribution function are special cases of the Fox–Wright function.{{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 | article-number = 061510 | doi = 10.1103/physreve.65.061510 | pmid = 12188735 | bibcode = 2002PhRvE..65f1510H | s2cid = 16276298 | 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 }} though nowadays the numeric computation can be done so efficiently{{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 | volume = 5 | issue = 4 | pages = 604–628 | doi = 10.3390/a5040604 | arxiv = 0911.4796 | s2cid = 15030084 | doi-access = free

History and further applications

::figure[src="https://upload.wikimedia.org/wikipedia/commons/4/43/Pibmasterplot.png" caption="doi=10.1021/jp025549u}} 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 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, 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.

Outside condensed matter physics, the stretched exponential has been used to describe the removal rates of small, stray bodies in the solar system,{{cite journal | author = Dobrovolskis, A., Alvarellos, J. and Lissauer, J. | year = 2007 | title = Lifetimes of small bodies in planetocentric (or heliocentric) orbits | journal = Icarus | volume = 188 | issue = 2 | pages = 481–505 | doi = 10.1016/j.icarus.2006.11.024 | bibcode = 2007Icar..188..481D }} the diffusion-weighted MRI signal in the brain,{{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}} and the production from unconventional gas wells.

In probability

If the integrated distribution is a stretched exponential, the normalized probability density function is given by p(\tau \mid \lambda, \beta)~d\tau = \frac{\lambda}{\Gamma(1 + \beta^{-1})} ~ e^{-(\tau \lambda)^\beta} ~ d\tau

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

Modified functions

A modified stretched exponential function f_\beta (t) = e^{ -t^{\beta(t)} } with a slowly t-dependent exponent β has been used for biological survival curves.{{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 | author = B. M. Weon | year = 2016 | title = Tyrannosaurs as long-lived species | journal = Scientific Reports | volume = 6 | article-number = 19554 | doi = 10.1038/srep19554 | pmid=26790747 | pmc=4726238| bibcode = 2016NatSR...619554W

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 I when the transmitters' locations are modeled as a 2D Poisson Point Process with no exclusion region around the receiver.{{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

The Laplace transform can be written for arbitrary fading distribution as follows: 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) where g is the power of the fading, \eta is the path loss exponent, \lambda is the density of the 2D Poisson Point Process, \Gamma(\cdot) is the Gamma function, and \mathbb{E}[x] is the expectation of the variable x.

The same reference also shows how to obtain the inverse Laplace Transform for the stretched exponential \exp\left(-s^\beta \right) for higher order integer \beta = \beta_q \beta_b from lower order integers \beta_a and \beta_b.

Internet streaming

The stretched exponential has been used to characterize Internet media accessing patterns, such as YouTube and other stable streaming media sites. The commonly agreed power-law accessing patterns of Web workloads mainly reflect text-based content Web workloads, such as daily updated news sites.

References

References

  1. (2021-06-14). "Slow stretched-exponential and fast compressed-exponential relaxation from local event dynamics". Journal of Physics: Condensed Matter.
  2. Holm, Sverre. (2020). "Time domain characterization of the Cole-Cole dielectric model". Journal of Electrical Bioimpedance.
  3. (2015). "Table of Integrals, Series, and Products". [[Academic Press, Inc.]].
  4. Dishon et al. 1985.
  5. (1 July 2002). "Friction on Small Objects and the Breakdown of Hydrodynamics in Solution: Rotation of Anthracene in Poly(isobutylene) from the Small-Molecule to Polymer Limits". The Journal of Physical Chemistry B.
  6. (2010-01-01). "A Better Way To Forecast Production From Unconventional Gas Wells". Society of Petroleum Engineers.
  7. Lei Guo, Enhua Tan, Songqing Chen, Zhen Xiao, and Xiaodong Zhang. (2008). ""The Stretched Exponential Distribution of Internet Media Access Patterns"".
  8. (2000). "Power-Law Distribution of the World Wide Web". Science.

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exponentials