Universal hyper-scaling relations, power-law tails, and data analysis for strong anomalous diffusion

TL;DR

This paper develops a universal theoretical framework for understanding strong anomalous diffusion, linking distribution tails, moment scaling, and crossover behavior, and provides practical methods for analyzing such diffusion in models.

Contribution

It introduces a novel asymptotic theory connecting distribution tails and moment exponents, offering explicit corrections and a robust scheme for parameter estimation in strong anomalous diffusion.

Findings

Derived relations between exponents $\xi$, $\zeta$, and $\alpha$

Provided explicit corrections to power-law behavior of moments

Validated theory with numerical and analytical results on five models

Abstract

Strong anomalous diffusion is {often} characterized by a piecewise-linear spectrum of the moments of displacement. The spectrum is characterized by slopes $\xi$ and $\zeta$ for small and large moments, respectively, and by the critical moment $\alpha$ of the crossover. The exponents $\xi$ and $\zeta$ characterize the asymptotic scaling of the bulk and 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 $\alpha$ to the corresponding power law. Our theory provides novel relations between 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. They provide the time scale that must be surpassed to clearly discriminate the leading-order power law from its sub-leading corrections. This insight allows us to point out sources of systematic errors in their numerical estimates. Rather than separately fitting an exponent for each moment we devise a robust scheme to determine $\xi$, $\zeta$ and $\alpha$. The findings are supported by numerical and analytical results on five different models exhibiting strong anomalous diffusion.