Log-periodic modulation in one-dimensional random walksL. Padilla, H. O. Mártin and J. L. Iguain
Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR) and Departamento de Física FCEyN, Universidad Nacional de Mar del Plata - Deán Funes 3350, (7600) Mar del Plata, Argentina
received 9 October 2008; accepted in final form 22 December 2008; published January 2009
published online 30 January 2009
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 due to the dependence of the diffusion coefficient on the length scale. Both the random walk exponent and the period of the modulation are analytically calculated and confirmed by Monte Carlo simulations.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
05.40.Fb - Random walks and Levy flights.
66.30.-h - Diffusion in solids.
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