Comb Model with Slow and Ultraslow Diffusion

¹ Max Planck Institute for the Physics of Complex Systems Nöthnitzer Strasse 38, 01187 Dresden, Germany
² Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia
³ Department of Physics, Technion, Haifa 32000, Israel
⁴ Institute for Physics and Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany
⁵ Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
⁶ Akhiezer Institute for Theoretical Physics, Kharkov 61108, Ukraine
⁷ Department of Physics and Astronomy, University of Padova, “Galileo Galilei” - DFA 35131 Padova, Italy

^⋆ Corresponding author. E-mail: chechkin@pks.mpg.de

Abstract

We consider a generalized diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyze the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.