Anomalous diffusion on a fractal mesh

TL;DR

This paper provides an exact analytical study of anomalous diffusion on a fractal mesh, revealing subdiffusive and superdiffusive behaviors influenced by fractal dimensions and memory effects.

Contribution

It introduces a novel analytical approach to model anomalous diffusion on a fractal mesh structure, incorporating a special construction algorithm and memory kernel effects.

Findings

Subdiffusion with $eta<1$ determined by fractal dimensions.

Superdiffusion observed with memory kernel influence.

Analytical expressions for dispersion relations obtained.

Abstract

An exact analytical analysis of anomalous diffusion on a fractal mesh is presented. The fractal mesh structure is a direct product of two fractal sets which belong to a main branch of backbones and side branch of fingers. The fractal sets of both backbones and fingers are constructed on the entire (infinite) $y$ and $x$ axises. To this end we suggested a special algorithm of this special construction. The transport properties of the fractal mesh is studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation $\langle x^2(t)\rangle\sim t^{\beta}$, where the transport exponent $\beta<1$ is determined by the fractal dimensions of both backbone and fingers. Superdiffusion with $\beta>1$ has been observed as well when the environment is controlled by means of a memory kernel.