EGUsphere

Copernicus Publications

Göttingen, Germany

10.5194/egusphere-2026-3498

A Hierarchical Fractional N-body Framework for Coulomb Stress Evolution in Fault Networks: Exact Analytical Solutions and Falsifiable Seismicity Scaling Laws

Chishtie

Farrukh A.

https://orcid.org/0000-0002-6392-6084

¹ ² ³

Peaceful Society, Science and Innovation Foundation, Vancouver, BC, Canada

Faculty of Land and Food Systems, University of British Columbia, Vancouver, BC, Canada

Department of Occupational Science and Occupational Therapy, University of British Columbia, Vancouver, BC, Canada

21 07 2026

2026 1 29

2026

This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/

This article is available from https://egusphere.copernicus.org/preprints/2026/egusphere-2026-3498/

The full text article is available as a PDF file from https://egusphere.copernicus.org/preprints/2026/egusphere-2026-3498/egusphere-2026-3498.pdf

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) &prop; 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.