kww

mdtools.functions.kww(t, tau=1, beta=1)[source]

Stretched exponential function, also known as Kohlrausch-Williams- Watts (KWW) function:

\[f(t) = \exp{\left[ -\left( \frac{t}{\tau} \right)^\beta \right]}\]

In physics, this function is often used as a phenomenological description of relaxation in disordered systems when the relaxation cannot be described by a simple exponential decay. Usually, it is then assumed that the system does not have a single relaxation mechanism with a single relaxation time, but the total relaxation is a superposition of multiple relaxation mechanisms with different relaxation times. The mean relaxation time is then given by

\[\langle \tau \rangle = \int_0^\infty \text{d}t \exp{\left[ -\left( \frac{t}{\tau} \right)^\beta \right]} = \frac{\tau}{\beta} \Gamma(\frac{1}{\beta})\]

with \(\Gamma(x)\) being the gamma function.

Parameters:

t (scalar or array_like) – Value(s) at which to evaluate \(f(t)\).
tau (scalar or array_like, optional) – Relaxation time(s).
beta (scalar or array_like, optional) – Stretching exponent(s).

Returns:

kww (scalar or numpy.ndarray) – The outcome of \(f(t)\). \(f(t)\) is evaluated element-wise if at least one of the input arguments is an array.