Fractional Moment Theory for Anomalous Transport: A Unified Framework for Lévy Flights, Fractals, and Complex Dynamical Systems

Fractional Moment Theory for Anomalous Transport: A Unified Framework for Lévy Flights, Fractals, and Complex Dynamical Systems | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 24 December 2025 V1 Latest version Share on Fractional Moment Theory for Anomalous Transport: A Unified Framework for Lévy Flights, Fractals, and Complex Dynamical Systems Author : Farrukh A. Chishtie 0000-0002-6392-6084 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176657959.91898288/v1 161 views 102 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We develop a unified mathematical framework extending classical moment theory from discrete integer orders to a continuous spectrum of real orders f> 0, providing systematic statistical characterization of complex systems exhibiting power-law behavior. This fractional moment theory addresses the fundamental problem in anomalous transport where traditional integer moments diverge for heavy-tailed distributions characteristic of Lévy flights, continuous time random walks, and chaotic advection. Through rigorous analysis of space-time fractional diffusion equations with Hilfer-composite time derivatives and Riesz-Feller space derivatives, we establish the operator-moment correspondence theorem proving that moments ⟨ | x | f ⟩ converge if and only if f, where α is the Lévy stability index governing asymptotic tail behavior u ( x ) ∼ | x | − ( 1 + α ) . We derive from first principles the universal scaling law ⟨ | x | f ⟩ = A f K µ, α f/α t µf/α with explicit coefficient formulas expressed through Gamma functions, establishing connections to Fox H-functions, Mittag-Leffler relaxation, and Wright functions. Complete proofs are provided using multiple independent methods including self-similarity analysis, Mellin transform techniques, and asymptotic expansions. Applications are developed for turbulent dispersion obeying Richardson’s four-thirds law, Lagrangian chaos characterized by finite-scale Lyapunov exponents, anomalous diffusion on fractal substrates, multifractal cascades, relaxation dynamics in glassy systems, epidemic spreading on scale-free networks, and extreme value distributions. The continuous parameter f enables extraction of scaling exponents and transport coefficients from systems where variance-based analysis fails entirely. Supplementary Material File (f-th moment framework chishtie mas 2025.pdf) Download 425.89 KB Information & Authors Information Version history V1 Version 1 24 December 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords anomalous diffusion complex systems fox h-function fractional calculus fractional moments lévy stable distributions mittag-leffler function richardson dispersion Authors Affiliations Farrukh A. Chishtie 0000-0002-6392-6084 [email protected] The University of British Columbia View all articles by this author Metrics & Citations Metrics Article Usage 161 views 102 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Farrukh A. Chishtie. Fractional Moment Theory for Anomalous Transport: A Unified Framework for Lévy Flights, Fractals, and Complex Dynamical Systems. Authorea . 24 December 2025. DOI: https://doi.org/10.22541/au.176657959.91898288/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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