Hybrid asymptotic-numerical approach for estimating first-passage-time densities of the two-dimensional narrow capture problem

A hybrid asymptotic-numerical method is presented for obtaining an asymptotic estimate for the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two-dimensional domain with a reflecting boundary. As motivation for this study, we calculate the variance in the capture time of a random walker by a single interior trap and determine this quantity to be comparable in magnitude to the mean. This implies that the mean is not necessarily reflective of typical capture times and that the full density must be determined. To solve the underlying diffusion equation, the method of Laplace transforms is used to obtain an elliptic problem of modified Helmholtz type. In the limit of vanishing trap sizes, each trap is represented as a Dirac point source that permits the solution of the transform equation to be represented as a superposition of Helmholtz Green's functions. Using this solution, we construct asymptotic short-time solutions of the first-passage-time density, which captures peaks associated with rapid capture by the absorbing traps. When numerical evaluation of the Helmholtz Green's function is employed followed by numerical inversion of the Laplace transform, the method reproduces the density for larger times. We demonstrate the accuracy of our solution technique with a comparison to statistics obtained from a time-dependent solution of the diffusion equation and discrete particle simulations. In particular, we demonstrate that the method is capable of capturing the multimodal behavior in the capture time density that arises when the traps are strategically arranged. The hybrid method presented can be applied to scenarios involving both arbitrary domains and trap shapes.

Figure 1

The full distribution of capture times $C\left(t\right)$ for a random walker in the unit disk, starting at $(0.3,0)$, and absorption at a circular trap of radius $\varepsilon =0.01$ centered at the origin.

Figure 2

Schematic and result for Example 3.1. In (a), we show the schematic with one trap of radius $\varepsilon =0.01$ centered at the origin (open circle), while the initial starting location for the particle is marked with a solid dot at ${\mathbf{x}}_0=(0.3,0)$. In (b), we show the exact capture time distribution computed from an eigenfunction solution of (1.1) (solid). A small time estimate obtained from the hybrid method is shown in dashed. Very close agreement is observed when $t$ is small.

Figure 3

Schematics of configurations for Examples 4.1 and 4.2. In each figure, the starting location ${\mathbf{x}}_0$ for the random walk is marked by a solid dot, while the nearest (farther) traps are indicated by heavy (light) solid circles. In both cases, there are $M=3$ nearest traps at a distance $\ell =0.4$ from ${\mathbf{x}}_0$. The traps have been enlarged for clarity.

Figure 4

Results for Examples 4.1 and 4.2. In (a), we show $P\left(t\right)$ for $M=3,\ell =0.4$, and $\varepsilon =0.01$ as numerically computed on the unit disk (solid) and square of side length 2 (dotted). The asymptotic estimate obtained from numerically inverting (4.5) is plotted by a dashed line. The three lines are almost indistinguishable. Details are given in the text. In (b), we show the corresponding plot for $C\left(t\right)$. As expected, the shape and size of the domain have little effect over the small time interval considered, so that (4.5) provides a good estimate for both the circular and square domains.

Figure 5

Schematic and results for Example 4.3. In (a), we show the schematic with one trap of radius $\varepsilon =0.01$ centered at ${\mathbf{x}}_1=r_1(cos{\theta }_1,sin{\theta }_1)$ with $r_1=0.6$ and ${\theta }_1=3\pi /4$ (open circle). The initial walker location (solid dot) is ${\mathbf{x}}_0=(0.7,0)$. In (b), we show the numerically computed capture time density (solid) along with the boundary-free approximation (dotted), one-term (dash-dotted), and two-term estimates (dashed). The two-term estimate is almost indistinguishable from the numerical solution of (1.1). Note that the two-term estimate is able to accurately predict the location of the mode of $C\left(t\right)$.

Figure 6

Results for Example 4.4. In (a), we show the free probability density $P\left(t\right)$ for a particle on a unit disk starting a random walk from ${\mathbf{x}}_0=(0.2,0)$. The locations of the five circular traps of radius $\varepsilon =0.01$ are given by (4.6). The full numerical result is shown by a solid line, the two-term asymptotic result by a dashed line, and the boundary-free approximation by a dotted line. In (b), we show the corresponding plots for $C\left(t\right)$. While both the boundary-free and two-term estimates are able to capture the mode of $C\left(t\right)$, the agreement of the two-term estimate persists well past the mode and into the tails of $P\left(t\right)$ and $C\left(t\right)$.

Figure 7

Schematic and results for Example 4.5. In (a), we show the schematic with ${\mathbf{x}}_0=(0.92,0)$ (solid dot) along with the locations of the five circular traps of radius $\varepsilon =0.01$ given by (4.21) (open circles). In (b) we show the full numerical result for $C\left(t\right)$ (solid), the two-term asymptotic estimate (dashed), and the boundary-free approximation (dotted). While the two-term estimate better predicts the full result over a longer time interval than does the boundary-free approximation, the interval of agreement is smaller than that seen in Fig. 6 due to the proximity of ${\mathbf{x}}_0$ to the boundary.

Figure 8

Schematic and results for Example 4.6. In (a), we show the starting location ${\mathbf{x}}_0$ at the origin (solid dot) surrounded by six traps of equal size that form two absorbing sets. The first set consists of the nearest trap centered at $(r_1,0)$. The second set is composed of the union of the five traps centered on a ring $r_c>r_1$. In (b), we show the resulting bimodal distribution for $C\left(t\right)$. The first peak corresponds to paths straight from ${\mathbf{x}}_0$ to $(r_1,0)$. The second peak corresponds to direct paths from ${\mathbf{x}}_0$ to the second absorbing set. The second absorbing set needs to be larger due to the greater dispersal of paths on the ring $r=r_c$ vs $r=r_1$.

Figure 9

Snapshots of solution $p(\mathbf{x},t)$ of (1.1) at (a) $t=0.02$ and (b) $t=0.05$ for Example 4.6. The times of the two snapshots approximately correspond to the times of the two peaks in $C\left(t\right)$ in Fig. 8. The first peak occurs as the initial distribution reaches the nearest trap, while the second peak occurs at a later time when the distribution spreads to the second set of five absorbing traps.

Figure 10

Schematic and results for Example 4.7. In (a), we show the starting location ${\mathbf{x}}_0$ at the origin (solid dot) surrounded by two absorbing sets. The first set consists of the nearest trap centered at $(r_1,0)$. The second set is composed of the union of the three traps centered on a ring $r_c>r_1$. In (b), we show the resulting bimodal distribution for $C\left(t\right)$. As in Example 4.6, the “wider net” cast by the second absorbing set accommodates the increased dispersal of paths by the time they have reached the ring, $r=r_c$.

Figure 11

Schematic and results for Example 4.8. In (a), we show the starting location ${\mathbf{x}}_0=(0.7,0)$ (solid dot) and a circular trap of radius $\varepsilon =0.01$ centered at ${\mathbf{x}}_1=(0,1/4)$. The elliptical (circular) domain is outlined by a heavy (light) solid line. In (b), we show the resulting distribution for $C\left(t\right)$, with the numerical result shown by a heavy solid line and the two-term estimate for the ellipse shown by a heavy dashed line. The light solid line is the two-term estimate for the same trap and starting locations, but with the ellipse replaced by the unit disk.

Figure 12

The full distribution of passage times for the configurations of Examples 3.1, 4.3, 4.6, and 4.7 over large times. Comparison given for full finite-element simulations (solid line) of (1.1), the hybrid-asymptotic method (dashed line), and discrete particle simulations (solid dots). At this scale, the hybrid result is very difficult to distinguish from the exact numerical result.

Figure 13

In (a), we show a schematic of a counterclockwise rotating trap of radius $\varepsilon =1\times {10}^{-5}$ starting at ${\mathbf{x}}_0=(0.5,0)$, while the starting location of the particles is at $(0.5cos\left(0.5\right),0.5sin\left(0.5\right))$. In (b), we show the capture time density $C\left(t\right)$. The peaks correspond to the capture of particles on each successive sweep of the trap through ${\mathbf{x}}_0$.