Parallel random target searches in a confined space

We study a random target searching performed by $N$ independent searchers in a $d$-dimensional domain of a large but finite volume. Considering the two initial distributions of searchers where searchers are either uniformly or point distributed, we estimate the mean time for the first of the searchers to reach the target and refer to it as searching time. The searching time for the uniformly distributed searchers exhibits a universal power-law dependence on $N$, irrespective of dimensionality and the target-to-domain size ratio. For point-distributed searching, the searching time has a logarithmic dependence on $N$ in the large $N$ limit, while in the small $N$ limit, it shows qualitatively different behaviors depending upon $r_0$, the initial distance of the searchers from a target. We obtain a diagram by comparing the searching times of the two initial distributions in the parameter space ($r_0,N$) and therein present the asymptotic lines separating three characteristic regions to explain numerical simulation results.

Figure 1

The searching times, defined as mean minimum FPT, for UPS as a function of rescaled searcher number, $N/N_1$. Symbols show the simulation data for various ratios of domain-to-target size, $b/a$, in $d$ dimension ($d=1,2,3$). All the simulation data coalesce to a single curve, displaying the universal behaviors in the small (solid) or large (dotted line) $N$ limit given by Eq. (23).

Figure 2

The minimum searching times of parallel searching performed by point-distributed searchers as functions of (rescaled) searcher number $Nq\left({\mathbf{r}}_0\right)$ for $d=2,3$ and $N$ for $d=1$: (a) when the initial position of the searcher is far from the target, $a\ll r_0\approx b$, and (b) when the searchers are initially close to the target, $a\approx r_0\ll b$. Simulation data (symbols) are in good agreement with the asymptotic behaviors given by Eq. (25) for the small $N$ limit (solid lines) and Eq. (31) for the large $N$ limit (dotted lines). The theoretical (solid) lines for small $N$ are presented only for $d=2,3$, because Eq. (25) is valid for $d\ge 2$ (see the text).

Figure 3

Schematic diagram of $〈t_{p,N}〉$ and $〈t_{u,N}〉$ as a function of $N$ when (a) $r_0\gg a$ or (b) $r_0\approx a$. $N_1$ is the crossover searcher number from the small $N$ to the large $N$ regime for UPS, while $N_2$ is for PPS. $N_3$ and $N_4$ correspond to the upper and lower boundaries of the region where PPS gives smaller searching times than UPS. (c) Schematic diagram indicating which parallel searching strategy performs better in parameter space of $r_0$ and $N$. Lines represent the analytic predictions on the domain boundaries, i.e., Eqs. (34, 35, 36).

Figure 4

Diagrams comparing $〈t_{u,N}〉$ and $〈t_{p,N}〉$, obtained from simulations, as a function of $N/N_1$ and $r_0/b$ for (a) $d=2$ and (b) $d=3$. The symbols represent three different regions where $〈t_{u,N}〉/〈t_{p,N}〉<1/\alpha$ (red crosses), $1/\alpha \le 〈t_{u,N}〉/〈t_{p,N}〉\le \alpha$ (black triangles), and $\alpha <〈t_{u,N}〉/〈t_{p,N}〉$ (blue circles) with $\alpha =3$. Lines indicate boundaries between different regions, analytically predicted based on the asymptotic expressions, i.e., using Eqs. (34, 35, 36). Small target size is realized by letting $b/a=2.5\times {10}^3$ for $d=2$ and $2.5\times {10}^2$ for $d=3$. The data are obtained from ${10}^1$–${10}^8$ ensembles, depending upon $N$, of Langevin dynamic simulations.