Universality in Anomalous Diffusion: Power Laws and Faster-Decaying Tails
Abstract
Strong anomalous diffusion is often characterized by a piecewise-linear spectrum of the moments of displacement. The spectrum is characterized by the slopes θ and ζ for small and large moments, respectively, and by the crossover moment qc at which the slope changes. The exponents θ and ζ characterize the asymptotic scaling of the bulk and of the tails of the probability distribution function of displacements, respectively. Here, we adopt asymptotic theory to match the behaviors at intermediate scales. The resulting constraint explains how distributions with algebraic tails imply strong anomalous diffusion, and it relates qc to the corresponding power law. Our theory provides universal hyper-scaling relations between the exponents characterizing strong anomalous diffusion, and it yields explicit expressions for the leading-order corrections to the asymptotic power-law behavior of the moments of displacement. We also characterize finite-time corrections and propose a robust distribution-based method to estimate θ, ζ, and qc. The findings are supported by numerical and analytical results for three paradigmatic models exhibiting strong anomalous diffusion. Finally, we show that power-law tails are not only sufficient but also necessary for this scenario: with a Gaussian, or generally fast-decaying, bulk and exponentially rare ballistic excursions, the spectrum of moments is single-linear, and the crossover to the apparent second branch drifts with the observation time. When the bulk is fast decaying and the tail is a power law, strong anomalous diffusion is still present, but the hyper-scaling relations among the exponents are violated.
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BibTeX
@misc{giberti2026universality,
title = {{Universality in Anomalous Diffusion: Power Laws and Faster-Decaying Tails}},
author = {Claudio Giberti and Alessandro Vezzani and Raffaella Burioni and Lamberto Rondoni},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05258},
eprint = {2611.05258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05258},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universality in Anomalous Diffusion: Power Laws and Faster-Decaying Tails
AU - Claudio Giberti
AU - Alessandro Vezzani
AU - Raffaella Burioni
AU - Lamberto Rondoni
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05258
UR - https://arxiv.org/abs/2611.05258
AB - Strong anomalous diffusion is often characterized by a piecewise-linear spectrum of the moments of displacement. The spectrum is characterized by the slopes θ and ζ for small and large moments, respectively, and by the crossover moment q_c at which the slope changes. The exponents θ and ζ characterize the asymptotic scaling of the bulk and of the tails of the probability distribution function of displacements, respectively. Here, we adopt asymptotic theory to match the behaviors at intermediate scales. The resulting constraint explains how distributions with algebraic tails imply strong anomalous diffusion, and it relates q_c to the corresponding power law. Our theory provides universal hyper-scaling relations between the exponents characterizing strong anomalous diffusion, and it yields explicit expressions for the leading-order corrections to the asymptotic power-law behavior of the moments of displacement. We also characterize finite-time corrections and propose a robust distribution-based method to estimate θ, ζ, and q_c. The findings are supported by numerical and analytical results for three paradigmatic models exhibiting strong anomalous diffusion. Finally, we show that power-law tails are not only sufficient but also necessary for this scenario: with a Gaussian, or generally fast-decaying, bulk and exponentially rare ballistic excursions, the spectrum of moments is single-linear, and the crossover to the apparent second branch drifts with the observation time. When the bulk is fast decaying and the tail is a power law, strong anomalous diffusion is still present, but the hyper-scaling relations among the exponents are violated.
ER -