Log-periodic modulation in one-dimensional random walks

TL;DR

This paper investigates how self-similar hopping rates in a one-dimensional lattice cause anomalous diffusion with mean-square displacement exhibiting power-law growth modulated by logarithmic periodic oscillations, confirmed through analysis and simulations.

Contribution

It analytically derives the origin and characteristics of log-periodic modulation in diffusion on self-similar lattices, supported by Monte Carlo simulations.

Findings

Mean-square displacement follows an anomalous power law.

Logarithmic periodic oscillations are observed and characterized.

Analytical calculations match simulation results.

Abstract

We have studied the diffusion of a single particle on a one-dimensional lattice. It is shown that, for a self-similar distribution of hopping rates, the time dependence of the mean-square displacement follows an anomalous power law modulated by logarithmic periodic oscillations. The origin of this modulation is traced to the dependence on the length of the diffusion coefficient. Both the random walk exponent and the period of the modulation are analytically calculated and confirmed by Monte Carlo simulations.