Anisotropic anomalous diffusion modulated by log-periodic oscillations

TL;DR

This paper studies how random walks on specially designed two-dimensional fractal-like structures exhibit direction-dependent anomalous diffusion, with oscillations modulating the diffusion behavior, and provides analytical and simulation results.

Contribution

It introduces a new class of anisotropic anomalous diffusion models on self-affine substrates with analytical solutions and simulation validation.

Findings

Different diffusion exponents in x and y directions.

Analytical expressions for exponents and oscillation periods.

Monte Carlo simulations confirm theoretical predictions.

Abstract

We introduce finite ramified self-affine substrates in two dimensions with a set of appropriate hopping rates between nearest-neighbor sites, where the diffusion of a single random walk presents an anomalous {\it anisotropic} behavior modulated by log-periodic oscillations. The anisotropy is revealed by two different random walk exponents, $\nu_x$ and $\nu_y$, in the {\it x} and {\it y} direction, respectively. The values of these exponents, as well as the period of the oscillation, are analytically obtained and confirmed by Monte Carlo simulations.