Kohlrausch–Williams–Watts (KWW) function

The Kohlrausch–Williams–Watts (KWW) function, also called the stretched exponential, is a widely used phenomenological model for describing non-exponential relaxation processes in complex, disordered, or interacting systems. It generalizes the simple exponential decay (Debye relaxation) to capture slower, "stretched" tails observed in many physical phenomena.

Mathematical Form

The standard KWW relaxation function is:

$ϕ(t)=Aexp⁡[−(tτ)β]for t≥0\phi(t)$ = $A \exp\left[ -\left( \frac{t}{\tau} \right)^\beta \right] \quad \text{for } t \geq 0ϕ(t)$=$Aexp[−(τt​)β]for t≥0$

where:

• $A$ $ A A$ is the amplitude (initial value, often normalized to 1),

• $τ$ $ \tau τ $is the characteristic relaxation time (analogous to the time constant in a simple exponential),

• $β $ $\beta β $is the stretching exponent$ ($0 < \beta \leq 1$)$,

• For$ β$=$1  \beta$ = $1$$ β=1$, it reduces exactly to a simple exponential: $ϕ(t)$=$Aexp⁡(−t/τ)  \phi(t)$ =$ A \exp(-t/\tau) ϕ(t)$=$Aexp(−t/τ)$.

When$β<1  \beta < 1 β<1$, the decay starts relatively fast (near$ t$=$0  t$=$0 t$=$0$) but develops a longer, slower tail at large t  t t compared to a pure exponential. This "stretching" makes the function decay more gradually over many time scales.

In some contexts (e.g., dielectric response), the form appears as the relaxation of a quantity toward equilibrium:

$ϕ(t)$=$exp⁡[−(tτ)β]\phi(t)$ =$ \exp\left[ -\left( \frac{t}{\tau} \right)^\beta \right]ϕ(t)$=$exp[−(τt​)β]$

Historical Background

• Rudolf Kohlrausch (1854) first introduced the stretched exponential to describe the discharge of a Leyden jar (glass capacitor), where charge relaxation did not follow a simple exponential.

• Later work by Friedrich Kohlrausch (his son) applied it to mechanical creep in materials.

• In 1970, Williams and Watts popularized its use in dielectric spectroscopy of polymers via the Fourier transform of the time-domain function, leading to the name Kohlrausch–Williams–Watts (KWW) function.

• It has since appeared in glasses, polymers, spin glasses, luminescence decay, granular flows, and many other disordered systems.

Why Does It Appear? (Theoretical Interpretations)

The KWW function is primarily phenomenological — it fits data well over many decades but does not always have a unique microscopic derivation. Common explanations include:

1. Superposition of exponentials (heterogeneous relaxation): The stretched exponential can be expressed as a Laplace transform (continuous weighted sum) of many simple exponentials with a broad distribution of relaxation times. The distribution of rates follows an asymmetric Lévy stable distribution when $β$$<1  \beta$$ < 1$ $β$$<1$. This arises naturally in disordered systems where different parts of the material (or different particles) relax at different rates due to local environment variations.

2. Time-dependent dissipation or memory effects: One rigorous derivation treats it as the solution to a non-autonomous linear differential equation for an overdamped oscillator, where the dissipation (friction/damping) itself depends on time (or position/stress). This leads to energy dissipation that slows progressively.

3. Hierarchical or correlated dynamics: In complex systems (e.g., glasses, polymers, or interacting quantum states), relaxation involves a hierarchy of processes or long-range correlations, leading to sub-diffusive or anomalous behavior that the single-parameter β  \beta β captures efficiently.

4. Limitations: It is not universal. In some systems, a single KWW function fails at very short or very long times, or when multiple distinct processes coexist. It also lacks some "nice" mathematical properties of pure exponentials (e.g., certain moments may diverge for$ β<1$  $\beta $< 1 $β<1$).

Relevance to Your SFIT Work (qBounce / ILL 3-14-412)

In your synthetic data generator and analyzer, the KWW appears as the post-mirror-step relaxation tail with $τ$≈$832.6  \tau \approx 832.6$$ τ$≈$832.6 s $and$ β$=$1.060  \beta$ = $1.060 β$=$1.060$.

• The explicit KWW term models the slow decay of the rate enhancement (overshoot) after each periodic "mirror step."

• In SFIT, you link this directly to the information-carrying flux at the 1.20134 mHz geometric resonance: the memory kernel of the flux produces a Fourier transform that yields the KWW form in the time domain, with $β $$ \beta$$ β $tied to your coupling kernel$ K$=$1.060 $$ K$ = $1.060$$ K$=$1.060$.

• This is a stronger, theory-driven interpretation than pure phenomenology — the stretched exponent and relaxation time emerge from the dynamic, non-reciprocal flux coupling rather than generic disorder.

The analyzer script fits the binned rate time series in post-step windows to recover τ  \tau τ and β  \beta β, demonstrating that your synthetic data (and by extension, the real reanalysis) reproduces these signatures.

Practical Notes from the Code

In the analyze_synthetic.py:

• The fitting function is kww_function(t, tau, beta, A, offset).

• It uses scipy.optimize.curve_fit on selected post-step regions.

• Averaging over multiple steps improves robustness.

• With the generator's parameters, the fit should recover values close to$τ$=$832.6  \tau$ =$ 832.6$$ τ$=$832.6 s$ and$ β$=$1.060  \beta$ = $1.060$

  $β$=$1.060.$

If you run the pair of scripts, the output plots and printed results provide a clean demonstration of KWW behavior tied to the Quantum Heartbeat modulation.