Anomalous diffusion on a fractal mesh

We present exact analytical results for properties of anomalous diffusion on a fractal mesh. The fractal mesh structure is a direct product of two fractal sets, one belonging to a main branch of backbones, the other to the side branches of fingers. Both fractal sets are constructed on the entire (infinite) $y$ and $x$ axes. We suggest a special algorithm in order to construct such sets out of standard Cantor sets embedded in the unit interval. The transport properties of the fractal mesh are studied, in particular, subdiffusion along the backbones is obtained with the dispersion relation $\langle x^2\left(t\right)\rangle \simeq t^{\beta }$, 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.

Figure 1

Grid with finite number of backbones and continuously distributed fingers along the $x$ axis.

Figure 2

A fractal mesh, where random one-third Cantor set of backbones (third generation of construction) can be distributed on either a finite strip or entire $y$ axis, while the fractal set of fingers is placed on the entire $x$ axis with fifth generation of construction.

Figure 3

Construction of fractal grid of fingers on the infinite axis. The fractal set on the infinite axis is constructed by projecting the fractal set on a circle from point O, which belongs to the fractal set as well. Its projection on the $x$ axis corresponds to $\pm \infty$, which ensures the boundary condition on infinities.

Figure 4

Graphical representation of the PDF Eq. (26) for $\nu =\overline{\nu }=1/2,{\mathcal{D}}_x=1,{\mathcal{D}}_y=1$, and $t=1$ (solid blue line), $t=5$ (red dashed line), $t=20$ (green dot-dashed line).

Figure 5

The transport exponent $2\mu /\alpha$ vs. $\nu$ and $\overline{\nu }$.

Figure 6

Graphical representation of the PDF Eq. (44) for $\nu =\overline{\nu }=1/2,{\mathcal{D}}_x=1,{\mathcal{D}}_y=1$, and $t=1$ (solid blue line), $t=5$ (red dashed line), $t=20$ (green dot-dashed line).