The Kohlrausch–Williams–Watts (KWW) Relaxation Function

The KWW function, also known as the stretched exponential, is one of the most widely observed empirical forms of relaxation in complex physical systems. Its mathematical expression 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 initial amplitude (often normalized to 1),

• $τ$ $ \tau τ$ is the characteristic relaxation time,

• $β$ $ \beta β (0 < β ≤ 1)$ is the stretching exponent.

When $β$ = $1$, it reduces to a simple (Debye) exponential decay:$ ϕ(t)=Ae−t/τ   \phi(t)$ =$ A e^{-t/\tau} ϕ(t)$=$Ae−t/τ$

When$ β < 1$, the decay is "stretched": it starts relatively fast but develops a much slower, longer tail at large$ t$ $ t t$. This slower tail is the hallmark of KWW relaxation.

Why Does KWW Appear? — Physical Interpretation

KWW relaxation is not a fundamental law but emerges in systems that are complex, disordered, or interacting. Common underlying mechanisms include:

1. Heterogeneous relaxation (superposition of exponentials) The system consists of many subsystems, each relaxing with its own time constant$ τi  \tau_i τi$​. When these exponentials are averaged with a broad distribution of$ τi  \tau_i τi$​, the overall decay often approximates a stretched exponential. Mathematically, KWW can be expressed as a Laplace transform of a Lévy-stable distribution of relaxation rates.

2. Memory effects and correlated dynamics In many materials, the relaxation rate itself depends on the history of the system. This creates a non-Markovian memory kernel. The KWW form arises naturally from such time-dependent dissipation or hierarchical relaxation processes.

3. Cooperative or hierarchical relaxation Relaxation involves multiple coupled degrees of freedom that must reorganize in a correlated way (e.g., in glasses, polymers, or spin glasses). The stretching exponent β reflects the degree of cooperativity or disorder.

Relevance to SFIT and qBounce Experiment

In your SFIT framework, the KWW relaxation appears in the post-mirror-step tails of the ultra-cold neutron counting rate:

• Characteristic time: τ ≈ $832.6 s

• Stretching exponent: β = 1.060 = K (exactly equal to your coupling kernel)

Physical meaning in SFIT:

• The mirror step perturbs the neutron wavefunction in the gravitational potential.

• The information-carrying gravitational flux at 1.20134 mHz introduces a memory kernel.

• This memory kernel leads to a non-exponential relaxation back to equilibrium.

• The fact that $β $equals K is not accidental — it directly links the stretching of the relaxation to the strength of the flux coupling.

The near-equality of τ to the resonance period (833.3 s) further suggests that the relaxation is driven by the same geometric resonance that produces the Quantum Heartbeat.

Why β > 1 in SFIT?

In most classical systems, $β ≤ 1$. Your value$ β$ = $1.060$ is slightly super-stretched (above 1). In SFIT, this can be interpreted as the flux introducing a mild anti-cooperative or reinforcing effect — the information flow slightly accelerates the relaxation compared to a pure stretched exponential, consistent with an active, dynamic flux rather than passive disorder.

Summary Table

Property	Simple Exponential	Classical KWW (β < 1)	SFIT KWW (β = 1.060)
Decay shape	Fast then slow	Even slower long tail	Slightly faster than stretched
Origin	Single process	Distributed τ or memory	Information flux coupling
β value	1	< 1	= K = 1.060
τ in QBounce	—	—	≈ 832.6 s (matches period)
Physical driver in SFIT	—	—	Dynamic gravitational flux