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

L. PADILLA; H. O. MÁRTIN; J. L. IGUAIN

PHYSICAL REVIEW E

AMER PHYSICAL SOC

Año: 2010 vol. 82 p. 11124 - 11130

1539-3755

Trabajo enviado: 15/04/2010; Aceptado:21/06/2010; Publicado:19/07/2010Under certain circumstances, the time behavior of a random walk is modulated by logarithmic periodic oscillations. Using heuristic arguments, we give a simple 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. The last example shows that the anomalous diffusion can arise from substrates without holes of all sizes.