Fractional Moment Theory for Anomalous Transport: A Unified Framework for Lévy Flights, Fractals, and Complex Dynamical Systems
Bibliographic record
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".