Stretched exponential function

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 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: == 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 ==

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. 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. 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. 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 ==