Stretched exponential function

Last updated 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 fit are also shown. 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

$f_{\beta }(t)=e^{-t^{\beta }}$ 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 β = 1, the usual exponential function is recovered. With a stretching exponentβ 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 β > 1) has less practical importance, with the notable exception of β = 2, which gives the normal distribution. In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. In mathematics, an exponential function is a function of the form In probability theory, the normaldistribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

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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 probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution. In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. The Fourier transform (FT) decomposes a function of time into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude component represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.

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;  therefore it is also called the Kohlrausch function. In 1970, G. Williams and D.C. Watts used the Fourier transform of the stretched exponential to describe dielectric spectra of polymers;  in this context, the stretched exponential or its Fourier transform are also called the Kohlrausch-Williams-Watts (KWW) function.

In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp(-t/τ). Rudolf Hermann Arndt Kohlrausch was a German physicist. Dielectric spectroscopy measures the dielectric properties of a medium as a function of frequency. It is based on the interaction of an external field with the electric dipole moment of the sample, often expressed by permittivity.

In phenomenological applications, it is often not clear whether the stretched exponential function should apply to 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     it can be shown that the asymptotic decay is a stretched exponential, but the prefactor is usually an unrelated power.

Mathematical properties

Moments

Following the usual physical interpretation, we interpret the function argument t as a time, and fβ(t) is the differential distribution. The area under the curve is therefore interpreted as a mean relaxation time. One finds

$\langle \tau \rangle \equiv \int _{0}^{\infty }dt\,e^{-(t/\tau _{K})^{\beta }}={\tau _{K} \over \beta }\Gamma \left({1 \over \beta }\right)$ where Γ is the gamma function. For exponential decay, 〈τ = τK is recovered. In mathematics, the gamma function is one of the extensions of the factorial function with its argument shifted down by 1, to real and complex numbers. Derived by Daniel Bernoulli, if n is a positive integer,

The higher moments of the stretched exponential function are: 

In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability, the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.

$\langle \tau ^{n}\rangle \equiv \int _{0}^{\infty }dt\,t^{n-1}\,e^{-(t/\tau _{K})^{\beta }}={{\tau _{K}}^{n} \over \beta }\Gamma \left({n \over \beta }\right).$ 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: 

$\rho (u)=-{1 \over \pi u}\sum \limits _{k=0}^{\infty }{(-1)^{k} \over k!}\sin(\pi \beta k)\Gamma (\beta k+1)u^{\beta k}$ 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 β = 1/2 where

$G(u)=u\rho (u)={1 \over 2{\sqrt {\pi }}}{\sqrt {u}}\exp(-u/4)$ 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 u = 1 as β approaches 1, corresponding to the simple exponential function.  Figure 2. Linear and log-log plots of the stretched exponential distribution function $G$ vs $t/\tau$ for values of the stretching parameter β between 0.1 and 0.9.

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. 

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.  For practical purposes, the Fourier transform may be approximated by the Havriliak-Negami function,  though nowadays the numeric computation can be done so efficiently  that there is no longer any reason not to use the Kohlrausch-Williams-Watts function in the frequency domain.

History and further applications

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,  the diffusion-weighted MRI signal in the brain,  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.

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.  

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