Stretched Exponential

📐 Definition

For $\tau > 0$ , $\alpha > 0$ , and $t \in \mathbb{R}$ ,

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

Domain and Codomain

Defined for real $t$ (or $t \ge 0$ in time-dependent settings). Output lies in $(0,1)$ for $t \neq 0$ and equals $1$ at $t=0$ .

⚙️ Key Properties

Monotone Decay and Scale

$f(t)$ decays monotonically in $|t|$ , with characteristic scale set by $\tau$ .

Derivative

f'(t) = -\frac{\alpha}{\tau^{\alpha}} |t|^{\alpha-1} \operatorname{sgn}(t)\, \exp\!\left[-\left(\frac{|t|}{\tau}\right)^{\alpha}\right]

🎯 Special Cases and Limits

$\alpha=1$ recovers ordinary exponential decay.
$\alpha=2$ yields a Gaussian in $t$ .
As $t\to\infty$ , $f(t)\to 0$ .
As $\alpha\to 0^+$ , $f(t)\to e^{-1}$ for $t\ne 0$ .

Stretched exponentials interpolate between exponential and Gaussian decay. They are closely connected to fractional dynamics, where memory kernels and anomalous relaxation often produce non-exponential tails.

Usage in Oakfield

Oakfield uses stretched-exponential forms as building blocks for fractional-time and colored-noise dynamics:

Subordination integrator: the subordination integrator uses a kernel of the form $w_\alpha(s)\propto \exp(-s^\alpha)$ in its quadrature rule when approximating subordinated evolution.
Stochastic noise: the stochastic_noise operator uses decays like $\exp(-(dt/\tau)^\alpha)$ when updating its internal (fractional OU–style) noise state, where $\alpha$ controls the “color”/memory.

Historical Foundations

📜 Kohlrausch Relaxation

Stretched-exponential relaxation laws appear in 19th-century work by Kohlrausch and were later widely used (often as the KWW form) to model non-exponential relaxation in complex media.

🌍 Modern Perspective

Stretched exponentials serve as flexible empirical decay models and as effective kernels in systems exhibiting heterogeneity or broad distributions of time scales.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §4
Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity