Logarithmic correction to fractal time behavior for diffusion in ultrametric spaces with random energy barriers

Abstract

A new model for the diffusion in ultrametric spaces with random transition rates is suggested. The ultrametric structure is generated by a branching process with a random number of branches, whereas the randomness of the transition rates is described in terms of an exponential distribution of activation energies. The interaction between these two random processes is analyzed. We show that the contribution of the ultrametric structure outweighs the contribution of the random energy barriers. The randomness of the energy barriers generates a logarithmic correction for large time. The distribution function ψ(t) of the waiting time in a given state has the following asymptotic behavior: ψ(t) ∼ t-1 · (ln vt)- (H + 1) Λ(ln ln vt) as t → ∞ where 1 ⩾ H > 0 is an exponent related to the ultrametric structure, v is the maximum jump frequency and Λ(ln ln vt) is a periodic function of ln ln vt.