Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics

We consider the survival probability $f\left(t\right)$ of a random walk with a constant hopping rate $w$ on a host lattice of fractal dimension $d$ and spectral dimension $d_s\le 2$, with spatially correlated traps. The traps form a sublattice with fractal dimension $d_a<d$ and are characterized by the absorption rate $w_a$ which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps ($w_a\ll w$), we find that $f\left(t\right)$ can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent $\alpha =1-(d-d_a)/d_w$, where $d_w$ is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power-law kinetics $f\left(t\right)\sim t^{-\alpha }$ with the same exponent $\alpha$ as for the stretched exponential regime. For strong absorption $w_a\gtrsim w$, including the limit of perfect traps $w_a\to \infty$, the stretched exponential regime is absent and the decay of $f\left(t\right)$ follows, after a short transient, the aforementioned power law for all times.

Figure 1

A log-log plot of the function $-ln\left[f\right(t\left)\right]$ for random walks on a line with a single trap for different values of the absorption probability $p_a$. For all curves, the initial position of the walker coincides with the position of the trap. Solid lines show the simulation results (averaged over about ${10}^7$ trajectories) and dashed lines show corresponding stretched exponential curves according to Eq. (28) with $c=0.56$.

Figure 2

A log-log plot of simulation data for the function $-ln\left[f\right(t\left)\right]$ for random walks on a line with absorption probability $p_a={10}^{-4}$ and different initial positions $i_0$. A single trap is located at the origin, $i=0$.

Figure 3

An iterative construction of a 1D lattice, with trap sites (highlighted by bold segments) forming a Cantor set. The first three generations $G_1,G_2$, and $G_3$ are shown. The cells of the host lattice are enumerated from left to right with nonnegative integers. Below each cell is its corresponding label in decimal (base 10) notation, and above is that same integer's ternary (base 3) representation. A site is a trap if and only if its ternary label does not contain the digit 1.

Figure 4

A log-log plot of the function $-ln\left[f\right(t\left)\right]$ for ${10}^7$ random walks on a $G_{20}$ Cantor lattice, for several values of the absorption probability $p_a$. Solid lines show the simulation results, dashed lines show the corresponding stretched exponential curves according to Eq. (20) with $\alpha =0.8155$, as given by (31), the coordination number $z=2$, and the empirical parameter $c=0.3$. Each trajectory begins from a randomly selected trap.

Figure 5

Composition and site enumeration schemes for the Sierpinski lattice. The first three generations $G_1, G_2$, and $G_3$ are shown. Traps, depicted as shaded elementary triangles, are characterized by ternary labels that do not contain the digit 1.

Figure 6

A log-log plot of the function $-ln\left[f\right(t\left)\right]$ for random walks on the Sierpinski lattice $G_{20}$ with a one-dimensional sublattice of traps (see Fig. 5) for different values of the absorption probability $p_a$. Solid lines show the simulation results (averaged over about ${10}^7$ trajectories), dashed lines show the corresponding stretched exponential curves according to (20) with $\alpha =0.748$, as given by (34), the coordination number of traps $z=3$, and the empirical constant $c=0.4$. Initial sites are chosen randomly from the trap lattice ${\mathcal{L}}_a$.

Figure 7

Recursive composition and cell enumeration of a 2D lattice with traps on the Sierpinski gasket (depicted by shaded cells). The first three generations $G_1,G_2,G_3$ are shown. Each cell is decorated with its binary address, which is the pair of its Cartesian coordinates $x$ (bottom) and $y$ (top), represented in base 2.

Figure 8

A log-log plot of the function $-ln\left[f\right(t\left)\right]$ for random walks on a 2D lattice with traps on the Sierpinski gasket $G_{20}$ (see Fig. 7) for different values of the absorption probability $p_a$. Solid lines show the simulation results (averaged over about ${10}^7$ trajectories), dashed lines show the corresponding stretched exponential curves according to (20) with $\alpha =0.79$, as given by (44), the coordination number $z=4$, and the empirical constant $c=0.2$. Initial sites are chosen randomly from among the trap sites.

Figure 9

A log-log plot of the survival probability $f\left(t\right)$ for random walks on the Sierpinski lattice $G_{30}$ with a one-dimensional sublattice of traps (see Fig. 5) for several values of the absorption probability $p_a$. Solid lines show simulation results (averaged over between ${10}^6$ and ${10}^8$ walks) and dashed lines show the corresponding power-law relaxation functions given by Eq. (47), $f\left(t\right)=c{p}_a^{-1}{\left(3wt\right)}^{-\alpha }$, with $\alpha =0.748$ and empirical constant $c=0.25$.

Figure 10

A log-log plot of the conditional probability $P\left(t\right)$ of surviving until, and occupying a trap at, time $t$ [see Eq. (11)]. It is evaluated over $N\sim {10}^8$ walks on the 2D lattice with imperfect traps on the Sierpinski gasket $G_{20}$ (see Fig. 7) for several values of the absorption probability $p_a$. Solid lines show simulation results, the dashed line shows the dependence (16), $P_0\left(t\right)=c{\left(wt\right)}^{-1+\alpha }$, and dash-dotted lines represent the ansatz $P\left(t\right)=\alpha {\gamma }^{-1}{\left(wt\right)}^{-1}$.