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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 teacher head, 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".