| 21 Jul 2026
A Hierarchical Fractional N-body Framework for Coulomb Stress Evolution in Fault Networks: Exact Analytical Solutions and Falsifiable Seismicity Scaling Laws
Abstract. We develop an exact analytical framework for the hierarchical fractional dynamics of interacting many-body systems and apply it to the nonlinear Coulomb stress transfer that governs seismicity in fractally organised fault networks. The framework rests on a scaling relation αk = 2 − 2 / (Nk + 1) that connects the order of a Riemann–Liouville fractional evolution equation at hierarchical level k to the number Nk of interacting bodies at that level, derived in a companion paper (Chishtie, 2026, Physics Open) and applied here to the seismogenic setting. Closed-form solutions are obtained via parametric trigonometric representations σk(θ) ∝ sin4 θ at each level together with Chebyshev polynomial inversions of the time–parameter relation, and they converge to the classical wave equation as Nk → ∞. We embed the framework into the Time-Dependent Stress Response (TDSR) seismicity model of Dahm and Hainzl (2022, J. Geophys. Res.) to describe the hierarchical Coulomb stress cascade through a fractally organised fault network. Three quantitative, falsifiable scaling laws follow from the closed-form solutions with no free parameters: an Omori–Utsu aftershock decay exponent pk = αk / 2 = 1 − 1 / (Nk + 1) that stratifies across generations; a spatial Coulomb stress falloff ℓ−(1+αk) that departs measurably from the classical elastic ℓ−3 law; and a Gutenberg–Richter b-value bk ≈Df Nk / (Nk + 1) that accounts for the longstanding discrepancy between the fractal-dimension expectation b = Df ≈ 1.5 – 2 and the commonly observed b ≈ 1 as a finite-Nk fractional correction. Numerical verification using the Grünwald–Letnikov scheme against the Mittag-Leffler exact series solution of the fractional relaxation equation confirms the analytical results. The standard TDSR model is recovered exactly as Nk → ∞ at all hierarchical levels and is therefore a special case of the present framework. We present illustrative qualitative comparisons with the 2023 Kahramanmaraş doublet and the 2024 Noto Peninsula earthquake; a systematic generation-stratified analysis across many sequences is set out as the primary observational test of the framework.
The manuscript requires a major revision for the following reasons:
1-The introduction requires a revision. Motivations and main objectives of the manuscript are not clear enough.
2-Definition 1 requires a rewritten.
3-Equation 10 requires a revision. Check dimensions please.
4-Equation 10 is not new. It has been treated largely in the literature.
5-Equation 11 is weird enough. Please check very carefully.
6-The confrontations with observations with observations.
7-Discuss the implications of fractional calculus in earthquakes. Check many missing references:
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Please revise.