Spatial Log Periodic Oscillations of First-Passage Observables in Fractals

TL;DR

This paper investigates how the mean first-passage time in fractal media exhibits both power-law scaling and log periodic oscillations, revealing detailed effects of fractal geometry on transport processes.

Contribution

It introduces a new expression for MFPT dependence on source-target distance that captures both power-law and log periodic oscillations in fractals.

Findings

MFPT scales with power law depending on fractal dimensions

Log periodic oscillations are observed in Sierpinski gasket data

Numerical results confirm the theoretical oscillatory behavior

Abstract

For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain a new expression for the source-target distance dependence of the MFPT that exhibits both the leading power law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log periodic oscillations that are a clear and definitive signal of the underlying fractal structure. We also present refined numerical results for the Sierpinski gasket that confirm this oscillatory behavior.