Log-periodic oscillations for diffusion on self-similar finitely ramified structures.
Under 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 basis 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 is fractal and the other is nonfractal, and confirmed by Monte Carlo simulations. The last example shows that the anomalous diffusion can arise from substrates without holes of all sizes.
Padilla
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Departamento de Física FCEyN, Instituto de Investigaciones Físicas de Mar del Plata, Universidad Nacional de Mar del Plata, Deán Funes 3350, 7600 Mar del Plata, Argentina.
Mártin
H O
HO
Iguain
J L
JL