Stretched exponential function

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Figure 1. Illustration of the stretched exponential fit. The data are rotational anisotropy of anthracene in polyisobutylene of several molecular masses. The plots have been made to overlap by dividing time (t) by the respective characteristic time constant.
Figure 1. Illustration of the stretched exponential fit. The data are rotational anisotropy of anthracene in polyisobutylene of several molecular masses. The plots have been made to overlap by dividing time (t) by the respective characteristic time constant.

The stretched exponential function, also known in the field of dielectric relaxation as the Kohlrausch-Williams-Watts (KWW) function, is a frequently used empirical description of the relaxation rates of many physical properties of complex systems such as polymers and glasses. It also appears to accurately describe the removal rates of small, stray bodies in the solar system. The function was introduced by the German physicist Rudolf Kohlrausch in 1854 to describe the discharge of a capacitor. Although not widely known, it was next used by A. Werner in 1907 to describe complex luminescence decays. It arises in the fluorescence decay law of electronic energy donors, as derived by Theodor Förster in 1949. The stretched exponential was again used by G. Williams and D.C. Watts in 1970 to characterize the dielectric relaxation rates in polymers. It is identical to the complementary cumulative Weibull distribution function.

The function is a simple extension of the exponential function with one additional parameter:

\phi (t) = e^{ - \left( {t /\tau _{WW} } \right)^\beta }

where \tau _{WW}\, is the characteristic relaxation time of the function and \beta\, is a parameter that can range between 0 and 1 and is referred to as the stretching parameter. Figure 1 shows the stretched exponential with the parameter of \beta\, equal to 0.52. For comparison, a least squares single and a double exponential fit are also shown. For another example see Figure 2 in the Lindsey and Patterson reference below.

[edit] Distribution function

A wide variety of relaxation behavior can be fit with the stretched exponential function, however, in most cases the fit is considered purely empirical, that is, it is used because it fits the data with a minimum number of parameters. It is possible, however, to ascribe some physical significance to the stretched exponential fit. In complex systems it may be reasonable to believe that the relaxation is intrinsically exponential, but that there is a large distribution of environments within the sample, each with different characteristics. The differences in the local environment leads to variations in the relaxation time, and when the experiment simultaneously measures a large ensemble of local relaxation times the result looks like a stretched exponential.

If the stretched exponential is the result of a distribution of relaxation times it is worthwhile to describe that distribution. If the distribution function of the stretched exponential is \rho_{WW}\,, then the following equation is correct:

e^{ - \left( {t /\tau _{WW} } \right)^\beta} = \int_0^\infty {e^{-t/\tau}\rho_{WW}(\tau)} \,d\tau

Lindsey and Patterson reported a formula for computing \rho_{WW}\,:

\rho_{WW} (\tau ) = - { {1} \over {\pi \tau }}\sum\limits_{k = 0}^\infty {{{( - 1)^k } \over {k!}}\sin (\pi \beta k)\Gamma (\beta k + 1)\left( {{\tau \over {\tau _{WW} }}} \right)} ^{\beta k }

For a more recent and general discussion, see Berberan-Santos et al., ref. given below. The distribution function G_{WW}\, plotted in Figure 2 is related to \rho_{WW}\, via the characteristic time constant \tau_{WW}\,:

G_{WW}=\tau_{WW}\rho_{WW} (\tau_{WW})\,

Figure 2 shows the same results plotted in both a linear and a log representation. The curves converge to a delta function at t / τWW = 1 as the stretching parameter \beta\, approaches one, corresponding to the simple exponential function.
Figure 2. Linear and log-log plots of the stretched exponential distribution function GWW vs t / τWW

for values of the stretching parameter β between 0.1 and 0.9.

[edit] Average and higher moments

In order to make valid comparisons between exponential decays and stretched exponential decays it is necessary to determine the meaning of the corresponding decay parameters. The decay parameter \tau\, of the exponential decay is the time necessary for decay amplitude to drop by a factor of e. However, this is not the case for stretched exponential decay parameter \tau _{WW}\, and so the decay parameters can not be compared directly. A more meaningful approach can be reached by noticing that the area under the curve of an exponential decay is proportional to the decay parameter.

\int_0^\infty{e^{-t/\tau}} \,dt =\tau

Thus, the two decay functions might be compared on the basis of their integrals. The integral of the stretched exponential is slightly more complex:

\int_0^\infty {e^{ - (t/\tau _{WW} )^\beta } } \,dt = {{\tau _{WW} } \over \beta }\Gamma ({1 \over \beta })

where Γ(x) is the gamma function, or generalized factorial. This follows immediately from Eq. 3.326 of Gradshteyn and Ryzhik. This allows us to define a common decay parameter for exponential and stretched exponential decay laws:

\tau = \begin{cases} \tau, & \mbox{exponential}\\ {{\tau _{WW} } \over \beta }\Gamma ({1 \over \beta }), & \mbox{stretched exponential} \end{cases}

The moments of the relaxation time can be found without explicit knowledge of the stretched exponential distribution function. It is necessary to show that:

\langle \tau ^n \rangle _{WW} = {1 \over {\Gamma (n)}}\int_0^\infty {t^{n - 1} \phi (t)dt}

is the nth moment of \tau _{WW}\, . Combining this equation with the expression for the stretched exponential in terms of the distribution function leads to:

\langle \tau^n \rangle _{WW} = {1 \over {\Gamma (n)}}\int_0^\infty {t^{n - 1} \left( {\int_0^\infty {e^{ - t/\tau} \rho (\tau)d\tau} } \right)dt}= {1 \over {\Gamma (n)}}\int_0^\infty {\rho (\tau)\left( {\int_0^\infty {t^{n - 1} e^{ - t/\tau } dt} } \right)d\tau}

where the order of integration has been changed to generate the last equallity.

\langle \tau ^n \rangle _{WW} = {1 \over {\Gamma (n)}}\int_0^\infty {\rho (\tau)\left( {\Gamma (n)\tau ^n } \right)d\tau}=\int_0^\infty {\tau ^n\rho (\tau)d\tau}

This last integral is the definition of the nth moment. The subscript has been omitted from \rho (\tau)\, to emphasize that it is not necessary to know the distribution function in this derivation. The final result is then:

\langle \tau^n \rangle _{WW} = {1 \over {\Gamma (n)}}\int_0^\infty {t^{n - 1} e^{ - (t/\tau _{WW})^\beta } dt}

Which can be converted to the final form using Gradshteyn and Ryzhik integral Eq. 3.478:

\langle \tau^n \rangle _{WW} ={{\tau _{WW} ^n } \over \beta }{{\Gamma (n/\beta )} \over {\Gamma (n)}}

[edit] References

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  • Kohlrausch, R. (1854). "Theorie des elektrischen Rückstandes in der Leidner Flasche". (Poggendorff) 91: 56–82, 179–213. Link
  • Williams, G. and Watts, D. C. (1970). "Non-Symmetrical Dielectric Relaxation Behavior Arising from a Simple Empirical Decay Function". Transactions of the Faraday Society 66: 80–85. doi:10.1039/tf9706600080.  DOI Link
  • Lindsey, C. P. and Patterson, G. D. (1980). "Detailed comparison of the Williams-Watts and Cole-Davidson functions". Journal of Chemical Physics 73: 3348–3357. doi:10.1063/1.440530.  DOI Link
  • Alvarez, F., Alegría, A. and Colmenero, J. (1991). "Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions". Physical Review B 44: 7306–7312. doi:10.1103/PhysRevB.44.7306.  DOI Link
  • I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, fourth edition. Academic Press, 1980. Errata.
  • Dobrovolskis, A., Alvarellos, J. and Lissauer, J. (2007). "Lifetimes of small bodies in planetocentric (or heliocentric) orbits". Icarus (journal) 188: 481–505. doi:10.1016/j.icarus.2006.11.024.  DOI Link
  • Baeurle, S.A., Hotta, A. and Gusev, A.A. (2005). "A new semi-phenomenological approach to predict the stress relaxation behavior of thermoplastic elastomers". Polymer (journal) 46: 4344–4354. doi:10.1016/j.polymer.2004.07.034.  DOI Link
  • Berberan-Santos, M.N., Bodunov, E.N. and Valeur, B. (2005). "Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential)". Chemical Physics 315: 171–182. doi:10.1016/j.chemphys.2005.04.006.  DOI Link
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