Hierarchical spatio-temporal coupling in fractional wanderings. (II). Diffusion phase diagram for Weierstrass walks

The one-dimensional Weierstrass walks (WW) model is developed in the framew ork of the extended (nonseparable) continuous-time random walk (CTRW) forma lism [1-12]. The WW model is a lacunary foundation of Levy walks [6-12] gen eralized to a nonconstant velocity. This nonconstant velocity is introduced by hierarchical, coherent spatio-temporal coupling adopted from the contin uous-time Weierstrass flights (CTWF) model developed in the previous paper [13]. Hence, for the probability density to pass by a walker in a single st ep, a random displacement with finite velocity is constructed as a geometri c series of the corresponding probability densities defined within a sequen ce of spatio-temporal scales. We calculated analytically and by Monte Carlo simulations the asymptotic in time mean-square displacement (MSD) of the w alker obtaining very good agreement between both approaches; also compariso n with corresponding results of the CTWF model is made. Considering differe nt types of the diffusion exponents, we constructed a diffusion phase diagr am on the plane defined by the spatial and temporal fractional dimensions w hich characterize our coupling. We obtained a diffusion exponent as a funct ion of these fractional dimensions covering all types of (nonbiased) diffus ion known up to now from the dispersive one over the normal, enhanced, ball istic, and hyperdiffusion up to the Richardson law of diffusion which defin es here a part of the borderline of the region where the MSD diverges. We o bserved that all kinds of anomalous diffusion are characterized by three ty pes of diffusion exponents only. For example, we found an asymptotic scalin g of MSD to occur with time for enhanced diffusion which was discovered by us within the CTWF model but is valid for a much more extended range of spa tio and temporal fractional dimensions. (C) 1999 Elsevier Science B.V. All rights reserved.