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 framework of the extended (nonseparable) continuous-time random walk (CTRW) formalism [1–12]. The WW model is a lacunary foundation of Lévy walks [6–12] generalized to a nonconstant velocity. This nonconstant velocity is introduced by hierarchical, coherent spatio-temporal coupling adopted from the continuous-time Weierstrass flights (CTWF) model developed in the previous paper [13]. Hence, for the probability density to pass by a walker in a single step, a random displacement with finite velocity is constructed as a geometric series of the corresponding probability densities defined within a sequence of spatio-temporal scales. We calculated analytically and by Monte Carlo simulations the asymptotic in time mean-square displacement (MSD) of the walker obtaining very good agreement between both approaches; also comparison with corresponding results of the CTWF model is made. Considering different types of the diffusion exponents, we constructed a diffusion phase diagram on the plane defined by the spatial and temporal fractional dimensions which characterize our coupling. We obtained a diffusion exponent as a function of these fractional dimensions covering all types of (nonbiased) diffusion known up to now from the dispersive one over the normal, enhanced, ballistic, and hyperdiffusion up to the Richardson law of diffusion which defines here a part of the borderline of the region where the MSD diverges. We observed that all kinds of anomalous diffusion are characterized by three types of diffusion exponents only. For example, we found an asymptotic scaling 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 spatio and temporal fractional dimensions.