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

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.

References in corpus (3)

• Critical Behavior of an Ising System on the Sierpinski Carpet: A Short-Time Dynamics Study
• On the occurrence of oscillatory modulations in the power-law behavior of dynamic and kinetic processes in fractals
• Log-periodic modulation in one-dimensional random walks

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• Local Einstein relation for fractals
• Universal behavior for single-file diffusion on a disordered fractal