Anomalous logarithmic slow-dynamics behavior on hierarchical and random systems

We study diffusion on hierarchical and random potential-well and barrier structures. In the case of a power-law distribution of barrier size for d=1 where the transition rate follows a Boltzmann distribution, we find ultraslow diffusion characterized by logarithmic time-dependent displacements. This result is valid for both barriers and wells. For d2 dimensions, we find a universal slow logarithmic dependence for the case of wells independent of d. For the case of barriers, we find that the logarithmic anomalous transport depends on the dimension of the system.