The stretched exponential function
is obtained by inserting a fractional
power law into the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
.
In most applications, it is meaningful only for arguments 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 exception of , which gives the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
.
In mathematics, the stretched exponential is also known as the
complementary cumulative Weibull 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 Maurice Re ...
. The stretched exponential is also the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
, 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
Rudolf Hermann Arndt Kohlrausch (November 6, 1809 in Göttingen – March 8, 1858 in Erlangen) was a German physicist.
Biography
He was a native of Göttingen, the son of the Royal Hanovarian director general of schools Friedrich Kohlrausch. He ...
in 1854 to describe the discharge of a capacitor; thus it is also known as 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 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, 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 time, and ''f''
β(''t'') is the differential distribution. The area under the curve
can thus be interpreted as a ''mean relaxation time''. One finds
where is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. For exponential decay, is recovered.
The higher
moments of the stretched exponential function are
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
Alternatively, a distribution
is used.
''ρ'' can be computed from the series expansion:
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
Figure 2 shows the same results plotted in both a
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and a
log
Log most often refers to:
* Trunk (botany), the stem and main wooden axis of a tree, called logs when cut
** Logging, cutting down trees for logs
** Firewood, logs used for fuel
** Lumber or timber, converted from wood logs
* Logarithm, in mathe ...
representation. The curves converge to a
Dirac delta function peaked at as ''β'' approaches 1, corresponding to the simple exponential function.
The moments of the original function can be expressed as
The first logarithmic moment of the distribution of simple-exponential relaxation times is
where Eu is the
Euler constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
.
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
In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function ''p'F'q''(''z'') based on ideas of and :
_p\ ...
. 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
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate cau ...
Rudolf Kohlrausch
Rudolf Hermann Arndt Kohlrausch (November 6, 1809 in Göttingen – March 8, 1858 in Erlangen) was a German physicist.
Biography
He was a native of Göttingen, the son of the Royal Hanovarian director general of schools Friedrich Kohlrausch. He ...
in 1854 to describe the discharge of a capacitor (
Leyden jar
A Leyden jar (or Leiden jar, or archaically, sometimes Kleistian jar) is an electrical component that stores a high-voltage electric charge (from an external source) between electrical conductors on the inside and outside of a glass jar. It typ ...
) 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
Theodor Förster (May 15, 1910 – May 20, 1974) was a German physical chemist known for theoretical work on light-matter interaction in molecular systems such as fluorescence and resonant energy transfer.
Education and career
Förster studied ...
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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
is given by
Note that confusingly some authors have been known to use the name "stretched exponential" to refer to the
Weibull 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 Maurice Re ...
.
Modified functions
A modified stretched exponential function
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
when the transmitters' locations are modeled as a 2D
Poisson Point Process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
with no exclusion region around the receiver.
The
Laplace Transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
can be written for arbitrary
fading distribution as follows:
where
is the power of the fading,
is the
path loss exponent,
is the density of the 2D Poisson Point Process,
is the Gamma function, and