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)...

The present study extends the correspondence principle of Martinez et al. that establishes a link between nonlinear Fokker–Planck equations (NLFPEs) and the variational principle approach of the theory of canonical ensembles. By virtue of the extended correspondence principle we reobtain...

The general approach of a nonlinear Fokker–Planck equation is applied to investigate the behavior of main statistical moments of a stochastic system. It was shown that the system described by Tsallis statistics can undergo transitions inherent to multiplicative noise-induced transitions. The...

To analyze the anomalous diffusion on a fractal structure with fractal in the time axis, we propose a statistical representation given by a path integral method in arbitrary fractal space-time. Using the method, we can understand easily several properties of the non-Gaussian-type behavior, and a...

We consider the diffusion of markers in a layered medium, with the lateral diffusion coefficient being the function of hight. We show that the probability density of the lateral displacements follows a one-dimensional Batchelor’s equation with time-dependent diffusion coefficient governed by...

We present a variety of models of random walk, discrete in space and time, suitable for simulating random variables whose probability density obeys a space–time fractional diffusion equation.

In the present paper, we consider the influence of weak dissipative effects on the passive scalar behavior in the framework of continuum percolation approach. The renormalization method of a small parameter in continuum percolation models is reviewed. It is shown that there is a characteristic...

The fractional diffusion equation is solved for different boundary value problems, these being absorbing and reflecting boundaries in half-space and in a box. Thereby, the method of images and the Fourier–Laplace transformation technique are employed. The separation of variables is studied for...