Log-periodic oscillations for diffusion on self-similar finitely ramified structures

TL;DR

This paper explains the origin of log-periodic oscillations in diffusion processes on self-similar, finitely ramified structures, supported by analytical derivations and Monte Carlo simulations.

Contribution

It provides a simple, pedagogical explanation for log-periodic oscillations in diffusion on self-similar, finitely ramified media, with analytical and simulation validation.

Findings

Analytical expressions for random walk exponent and oscillation period.

Confirmation of theoretical predictions through Monte Carlo simulations.

Identification of hierarchical diffusion constants as the origin of oscillations.

Abstract

Under certain circumstances, the time behavior of a random walk is modulated by logarithmic periodic oscillations. The goal of this paper is to present a simple and pedagogical explanation of the origin of this modulation for diffusion on a substrate with two properties: self-similarity and finite ramification order. On these media, the time dependence of the mean-square displacement shows log-periodic modulations around a leading power law, which can be understood on the base of a hierarchical set of diffusion constants. Both the random walk exponent and the period of oscillations are analytically obtained for a pair of examples, one fractal, the other non-fractal, and confirmed by Monte Carlo simulations.