Residence time of symmetric random walkers in a strip with large reflective obstacles

We study the effect of a large obstacle on the so-called residence time, i.e., the time that a particle performing a symmetric random walk in a rectangular (two-dimensional, 2D) domain needs to cross the strip. We observe complex behavior: We find out that the residence time does not depend monotonically on the geometric properties of the obstacle, such as its width, length, and position. In some cases, due to the presence of the obstacle, the mean residence time is shorter with respect to the one measured for the obstacle-free strip. We explain the residence time behavior by developing a one-dimensional (1D) analog of the 2D model where the role of the obstacle is played by two defect sites having smaller probability to be crossed with respect to all the other regular sites. The 1D and 2D models behave similarly, but in the 1D case we are able to compute exactly the residence time, finding a perfect match with the Monte Carlo simulations.

Figure 1

Representation of the lattice $\mathrm{\Lambda }$. The circles denote the obstacle made of sites which cannot be accessed by the particle. The parameters $H$ and $W$ are, respectively, the height and the width of the obstacle. Bold arrows denote a trajectory of the 2D random walk entering the lattice from the left and exiting it from the right.

Figure 2

Residence time vs obstacle height. The obstacle is placed at the center of the strip and its width is $W=2$ (disks), $W=20$ (triangles), $W=40$ (squares), and $W=60$ (diamonds). Simulation parameters: $L_1=200$, $L_2=50$, and total number of inserted particles $5\times {10}^7$. The total number of particles exiting through the right boundary decreases when the obstacle height is increased from $2.49\times {10}^5$ (empty strip) to $1.69\times {10}^5$ (disks), $0.99\times {10}^5$ (triangles), and $0.68\times {10}^5$ (squares), and $0.52\times {10}^5$ (diamonds) for $H=46$. The dashed line at 26 930 represents the value of the residence time measured for the empty strip.

Figure 3

Residence time vs obstacle width. The obstacle is placed at the center of the strip and its height is $H=40$. The total number of particles exiting through the right boundary decreases when the obstacle width is increased from $2\times {10}^5$ for $W=2$ to $0.5\times {10}^5$ for $W=198$. The parameters of the simulation and the dashed line are as in Fig. 2.

Figure 4

Residence time vs squared obstacle side length. The squared obstacle is placed in the middle of the strip. The total number of particles exiting through the right boundary decreases when the obstacle side length is increased from $2.49\times {10}^5$ (empty strip) to $0.63\times {10}^5$ for side length equal to 46. The parameters of the simulation and the dashed line are as in Fig. 2.

Figure 5

Residence time vs integer part of the abscissa (first coordinate) of the geometric center of the obstacle. Disks refer to a squared obstacle with side length 40, whereas triangles refer to a rectangular obstacle with width $W=2$ and height $H=40$. The total number of particles exiting through the right boundary is approximately $1.24\times {10}^5$ (disks) and $2.01\times {10}^5$ (triangles) and depends poorly on the obstacle position. The parameters of the simulation and the dashed line are as in Fig. 2.

Figure 6

Mean time spent by the walker crossing the strip in each site of the strip (local residence time) for the empty strip case (black) and in the presence of the obstacle (gray). Data are those of the experiment described by the disks in Fig. 5. The obstacle is a squared obstacles with side length 40 whose center is placed at the site with integer part of the abscissa equal to 60.

Figure 7

Total residence time in the left (circles), central (squares), and right (triangles) parts of the strip in the presence of the obstacle (gray) and in the empty strip case (black). The experiment associated with the disks in Fig. 5 is considered.

Figure 8

Representation of the lattice on which the 1D model is defined. Open and solid circles respectively represent regular and special (defect) sites. The two squared boxes represent the two boundary absorbing sites. Recall that $L=n+h+w+3$.

Figure 9

Residence time vs $p$. Simulation parameters: total number of inserted particles $5\times {10}^7$, $L=201$, $n=98$ and $h=2$ (disks), $n=89$ and $h=20$ (triangles), $n=79$ and $h=40$ (squares), and $n=69$ and $h=60$ (diamonds). The total number of particles exiting through the right exit decreases when $p$ grows from $2.48\times {10}^5$ (no defect is present, namely, $p=0$) to $1\times {10}^5$ (disks), $0.22\times {10}^5$ (triangles), $0.12\times {10}^5$ (squares), and $0.08\times {10}^5$ (diamonds) for $p=0.99$. The dashed line at 13 466 represents the value of the residence time measured in absence of defect sites ($p=0$). The solid line is the exact solution.

Figure 10

Residence time vs $h$ (even) for $p=0.84$, $L=201$, $n=(L-h-3)/2$, and total number of inserted particles $5\times {10}^7$. The dashed and the solid lines are as in Fig. 9. The total number of particles exiting through the right exit decreases when $h$ is increased from $2.31\times {10}^5$ for $h=2$ to $0.41\times {10}^5$ for $h=194$.

Figure 11

Residence time vs $n$ for $p=0.84$, $L=201$, $h=40$ (disks), $h=2$ (triangles), and total number of inserted particles $5\times {10}^7$. The dashed and the solid lines are as in Fig. 9. The total number of particles exiting through the right exit is approximately equal to $1.2\times {10}^5$ (disks) and $2.3\times {10}^5$ (triangles).

Figure 12

Residence time as a function of $h$ and $p$, for $L=201$ and defect sites symmetric with respect to the middle point of the lane. For $p=0$, the residence time is 13 466 (absence of defect sites).

Figure 13

Residence time as a function of $n$ and $h$, for $L=201$ and $p=0.9$. Recall that in the case of symmetric random walk (no defect sites), the residence time is 13 466.

Figure 14

Total residence time in the left (circles), central (squares), and right (triangles) regions of the lane in the presence of the defect sites (gray) and in the $p=0$ case (black). The experiment associated with the disks in Fig. 11 is considered.

Figure 15

Schematic representation of the five-state chain model.

Figure 16

Residence time vs $L$ for the lane with no singular sites (symmetric case). The solid line is the exact solution (4.11), whereas circles are the average results of a Monte Carlo simulation with $5\times {10}^8$ particles started at the site 1.