Anomalous diffusion as a stochastic component in the dynamics of complex processes

TL;DR

This paper introduces a method using the second-order difference moment to identify and analyze anomalous diffusion in various complex natural processes, demonstrating its effectiveness across diverse examples.

Contribution

It presents a novel interpolation approach based on the difference moment for detecting and characterizing anomalous diffusion in steady-state complex systems.

Findings

Successfully applied to quantum dot fluorescence, X-ray emission, fluid velocity, and geoelectrical signals.

Adequately describes stochastic behavior in all tested complex processes.

Broadens the scope of processes where anomalous diffusion can be identified.

Abstract

We propose an interpolation expression using the difference moment (Kolmogorov transient structural function) of the second order as the average characteristic of displacements for identifying the anomalous diffusion in complex processes when the stochastic dynamics of the system under study reaches a steady state (large time intervals). Our procedure based on this expression for identifying anomalous diffusion and calculating its parameters in complex processes is applied to the analysis of the dynamics of blinking fluorescence of quantum dots, X-ray emission from accreting objects, fluid velocity in Rayleigh-B\'enard convection, and geoelectrical signal for a seismic area. For all four examples, the proposed interpolation is able to adequately describe the stochastic part of the experimental difference moment, which implies that anomalous diffusion manifests itself in these complex processes. The results of this study make it possible to broaden the range of complex natural processes in which anomalous diffusion can be identified.