Diffusive process on a backbone structure with drift terms

The effects of an external force on a diffusive process subjected to a backbone structure are investigated by considering the system governed by a Fokker-Planck equation with drift terms. Our results show an anomalous spreading which may present different diffusive regimes connected to anomalous diffusion and stationary states.

Figure 1

Illustration of the backbone structure used in the comb model. Note that diffusion in the $x$ direction only occurs when $y=0$ and diffusion in the $y$ direction is perpendicular to the $x$ axis.

Figure 2

Time behavior of the mean square displacement with $v_x=0$, ${\mathcal{D}}_y=20$, ${\mathcal{D}}_x=10$, and $\tilde{y}=0.8$. The solid (black) line corresponds to the case $v_y=0.1$ and the dashed (red) line is the case $v_y=1$. Note that saturation in the $x$ direction is reached more rapidly for large values of $v_y$ and more slowly for small values of $v_y$. The dotted (blue) line was incorporated in order to illustrate the subdiffusive behavior presented by the system before reaching the stationary state.

Figure 3

Time behavior of the mean square displacement for $v_y=0$, ${\mathcal{D}}_y=20$, ${\mathcal{D}}_x=10$, and $\tilde{y}=0.8$. The straight dotted lines were added in order to illustrate the usual and the subdiffusive behavior which can be manifested by the mean square displacement obtained from Eqs. (8) and (9). Dashed (blue), solid (red), and dotted (green) lines correspond to the cases ${\overline{v}}_x=10$, ${\overline{v}}_x=1$, and ${\overline{v}}_x=0.1$, respectively, as indicated.

Figure 4

Time behavior of the mean square displacement for $v_y=5\times {10}^{-4}$, $v_x=0$, ${\overline{v}}_x=1$, ${\mathcal{D}}_y=5$, ${\mathcal{D}}_x=10$, and $\tilde{y}=0.1$. The dotted (blue) and dashed (red) lines were incorporated in order to illustrate the subdiffusive behaviors presented by the system before reaching the stationary state.