Recently, a modified fractional diffusion equation has been proposed [I. Sokolov, J. Klafter, From diffusion to anomalous diffusion: a century after Einstein's brownian motion, Chaos 15 (2005) 026103; A.V. Chechkin, R. Gorenflo, I.M. Sokolov, V.Yu. Gonchar, Distributed order time fractional diffusion...

We show that non-linear diffusion equations can describe state-dependent diffusion, i.e., fission–fusion dynamics. We thereby provide a new dynamical basis for understanding Tsallis distributions (q-Gaussian distributions), anomalous diffusion (subdiffusion, superdiffusion and superballistic...

Exact solutions are rare for non-Markovian random walk models even in 1D, and much more so in 2D. Here we propose a 2D genuinely non-Markovian random walk model with a very rich phase diagram, such that the motion in each dimension can belong to one of 3 categories: (i) subdiffusive, (ii)...

Modeling the rate of nucleotide substitutions in DNA as a dichotomous stochastic process with an inverse power-law correlation function describes evolution by a fractal stochastic process (FSP). This FSP model agrees with recent findings on the relationship between the variance and mean number...

Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those equations, the explicit laws of certain stable processes turn out to be...

The anisotropic sub-diffusion random walks on multi-dimensional comb structure model have been studied in the continuum approximation. The problem is that in the considered model mean square displacements on different directions have different power temporal dependencies. So this case...

We propose and study an analytic model for growing interfaces in the presence of Brownian diffusion and hopping transport. The model is based on a continuum formulation of mass conservation at the interface, including reactions. The Burgers-KPZ equation for the rate of elevation change emerges...

The fractional Fokker–Planck equation, were used to describe the anomalous diffusion in external fields, is derived using a comb-like structure as a background model. For the force-free case, the distribution function associated with space dependence diffusion coefficient along the backbone of...

Investigations on diffusion in systems with memory [I.V.L. Costa, R. Morgado, M.V.B.T. Lima, F.A. Oliveira, Europhys. Lett. 63 (2003) 173] have established a hierarchical connection between mixing, ergodicity, and the fluctuation–dissipation theorem (FDT). This hierarchy means that ergodicity...

Einstein's theory of Brownian motion is revisited in order to formulate a generalized kinetic theory of anomalous diffusion. It is shown that if the assumptions of analyticity and the existence of the second moment of the displacement distribution are relaxed, the fractional derivative naturally...