The Monad‑Field: Comparative Relaxation Signatures under a Nonlinear Magnetar‑Like Mapping
White Paper: Comparative Relaxation Signatures under a Nonlinear Magnetar‑Like Mapping
Abstract
This study compares two relaxation models—an empirical stretched‑exponential form and a constant‑order fractional (Mittag–Leffler) model—subjected to a shared nonlinear observational mapping motivated by magnetar timing phenomenology. The mapping applies a time‑dilation factor (1 + k·S(t)), cumulative phase integration, and extraction of the lag‑rate envelope. An effective relaxation exponent β_eff is obtained by fitting the late‑time decay of the lag‑rate to a stretched‑exponential surrogate.
The resulting β_eff(k) curves exhibit qualitatively distinct behaviours. Stretched‑exponential substrates show strong and opposite coupling trends for β_true = 0.5 and β_true = 0.2. In contrast, constant‑order fractional substrates yield β_eff values that are nearly independent of coupling. These findings indicate that the nonlinear mapping produces distinguishable phenomenological signatures for the two relaxation laws considered.
1. Introduction
Timing residuals from highly stable astrophysical rotators can be influenced by an underlying relaxation process. We adopt a simplified model in which the instantaneous angular velocity is modulated by a dimensionless relaxation function S(t):
dθ_eff/dt = Ω / (1 + k·S(t)),
where Ω is the unperturbed spin rate and k is a coupling parameter. The cumulative phase lag is
lag(t) = (θ_eff(t) − Ωt) / (2π),
and its time derivative provides the observable decay envelope. An effective exponent β_eff is extracted by fitting the late‑time decay of |laġ(t)| to a stretched‑exponential form.
2. Relaxation Models
2.1 Stretched‑Exponential Substrates
We consider the empirical form:
S(t) = exp[−(t/τ)^{β_true}], τ = 0.5 s,
with two representative exponents:
- β_true = 0.5 (fluidic / impedance‑like)
- β_true = 0.2 (topological / phase‑gradient‑like)
2.2 Fractional (Mittag–Leffler) Substrates
We solve the constant‑order fractional relaxation equation:
Dᵗᵅ S(t) = −τ^(−α) S(t), S(0) = 1,
using a Grünwald–Letnikov discretisation of the Caputo derivative. The solution is the Mittag–Leffler function:
S(t) = Eₐ[−(t/τ)ᵅ],
which exhibits late‑time algebraic decay S(t) ∼ t^(−α). We consider α = 0.5 and α = 0.2 to parallel the stretched‑exponential cases. Simulations use dt = 0.002 s over a 30 s window.
3. Extraction of β_eff
For each substrate S(t) and coupling k ∈ {1, 2, 5, 10, 15, 20}, we:
- Compute the lag and its derivative.
- Identify the peak of |laġ(t)|.
- Exclude the first 0.5 s after the peak to remove the startup transient.
- Fit the remaining decay to A·exp[−(t/τ)^{β_eff}].
The fitted β_eff is used as a phenomenological descriptor of the decay shape under the mapping.
4. Results
4.1 Constitutive Fingerprint Matrix
| Model | k=1 | k=2 | k=5 | k=10 | k=15 | k=20 |
|---|---|---|---|---|---|---|
| Fluidic (β_true = 0.5) – Analytic | 0.597 | 0.669 | 0.814 | 0.958 | 1.000 | 1.000 |
| Fluidic (α = 0.5) – GL (Mittag–Leffler) | 0.287 | 0.286 | 0.285 | 0.283 | 0.281 | 0.279 |
| Topological (β_true = 0.2) – Analytic | 0.253 | 0.273 | 0.193 | 0.130 | 0.098 | 0.079 |
| Topological (α = 0.2) – GL (Mittag–Leffler) | 0.259 | 0.259 | 0.258 | 0.258 | 0.258 | 0.258 |
5. Interpretation
5.1 Stretched‑Exponential Substrates
The analytic models exhibit strong coupling dependence. For β_true = 0.5, β_eff increases toward unity. For β_true = 0.2, β_eff decreases markedly with k. These opposite trends arise from the interaction between the intrinsic decay law and the nonlinear mapping.
5.2 Fractional Substrates
The constant‑order fractional models yield β_eff values that vary only weakly with coupling. This reflects the dominance of the algebraic tail in the fitted region and the limited sensitivity of the stretched‑exponential surrogate to differences in fractional order. In particular, β_eff for α = 0.2 remains stable at approximately 0.258 across the entire coupling range.
5.3 Caveats
- β_eff is a phenomenological parameter and does not recover the intrinsic fractional order α.
- The extraction pipeline compresses different fractional orders into a narrow β_eff range (≈0.26–0.29).
- The comparison is specific to the adopted mapping and fitting protocol.
- No stochastic forcing or uncertainty quantification is included; results are deterministic.
6. Conclusion
Under the adopted nonlinear mapping, stretched‑exponential and constant‑order fractional relaxation models produce distinct β_eff(k) signatures. The stretched‑exponential substrates show strong and opposite coupling trends, while the fractional substrates exhibit near‑invariance. These results provide a quantitative reference for distinguishing phenomenological relaxation models from fractional‑memory ones in simulated timing data.
