Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

The stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed. The particle is subjected to a stochastic stimulation obeying a general Lévy stable statistics and experiences long rests due to nonhomogeneously distributed traps. The memory is taken into account by subordination of that process to a random time; then the subordination equation is position dependent. The problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power-law position dependence of the memory. In the case of the Gaussian statistics, the density distribution and moments are derived: depending on geometry and memory parameters, the system may reveal both the subdiffusion and enhanced diffusion. The similar analysis is performed for the Lévy flights where the finiteness of the variance follows from a variable noise intensity near a boundary. Two diffusion regimes are found: in the bulk and near the surface. The anomalous diffusion exponent as a function of the system parameters is derived.

Figure 1

Distributions obtained from a numerical integration of Eq. (21) with $g\left(x\right)={\left|x\right|}^{\theta \alpha }$ and $f\left(x\right)={\left|x\right|}^{-\gamma }$ for $\alpha =3/2$, $\theta =2/3$, $\beta =1/2$, $\gamma =1/2$, and $t=0.1$. Curves correspond to the following values of ${\lambda }_I$: 0 (magenta), 0.1 (blue), 0.2 (cyan), 0.5 (red), and 1 (green) (from bottom to top in the upper part and from top to bottom in the lower part). The straight lines correspond to a power-law function with the index 2.5, 2.9, and 3.25 (from top to bottom).

Figure 2

Variance as a function of time for $\alpha =1$ and $\beta =1/2$. The curves (from bottom to top) correspond to (1) $\gamma =4$ and $L=10$; (2) $\gamma =2$ and $L=10$; (3) $\gamma =2$ and $L=100$. Straight-line segments (marked by the red lines) on the left-hand side have the slope 1/2 and those on the right-hand side have 1/5, 1/3, and 1/3.

Figure 3

Variance as a function of time for a few sets of the parameters $\alpha$, $\theta$, and $\beta$; the other parameters are $L=10$ and $\gamma =2$.

Figure 4

Slope of the time dependence of the variance $t^{\mu }$ as a function of $\theta$ for $\beta =0.5$ and two values of $\alpha$: 0.5 (lower points) and 1.5 (upper points). The other parameters are $\gamma =2$ and $L=10$. Solid lines mark the function $\mu =0.5/[1+c(\alpha \left)\theta \right]$, where $c\left(0.5\right)=1.51$ and $c\left(1.5\right)=0.51$.

Figure 5

Slope as a function of $\alpha$ for two values of $\theta$, the other parameters are the same as in Fig. 4. Solid line marks the function $\mu =0.2+0.12\alpha$.

Figure 6

Slope as a function of $\beta$ for two values of $\theta$, the other parameters are the same as in Fig. 4. Solid lines mark the functions $\mu =0.09+0.545\beta$ and $\mu =-0.05+1.276\beta$ for the positive and negative $\theta$, respectively.