Fastest first-passage time for multiple searchers with finite speed

Abstract

We study analytically and numerically the mean fastest first-passage time (fFPT) to an immobile target for an ensemble of $N$ independent finite-speed random searchers driven by dichotomous noise and described by the telegrapher's equation. In stark contrast to the well-studied case of Brownian particles -- for which the mean fFPT vanishes logarithmically with $N$ -- we uncover that the mean fFPT is bounded from below by the minimal ballistic travel time, with an exponentially fast convergence to this bound as $N \to \infty$. This behavior reveals a dramatic efficiency advantage of physically realistic, finite-speed searchers over Brownian ones and illustrates how diffusive macroscopic models may be conceptually misleading in predicting the short-time behavior of a physical system. We extend our analysis to anomalous diffusion generated by Riemann-Liouville-type dichotomous noises and find that target detection is more efficient in the superdiffusive regime, followed by normal and then subdiffusive regimes, in agreement with physical intuition and contrary to earlier predictions.

Figures (7)

• Figure 1: (Color online) 3D plot of $B_{N,\gamma}$ obtained from numerical evaluation of the integrals in Eqs. \ref{['eq:B_def']} and \ref{['eq:f_def']}, as function of $N$ and $\gamma$. The cyan solid line indicates our asymptotic large-$\gamma$ prediction $2\gamma/(\pi\ln N)$, see Eqs. \ref{['inter']} and \ref{['z']}.
• Figure 2: (Color online) Excess factor $B_{N,\gamma}$ for $\gamma=5$ (blue), $\gamma=10$ (green) and $\gamma=15$ (red) as function of $N$. Colored solid, dashed and dot-dashed curves depict the corresponding factors $B_{N,\gamma}$ obtained by a numerical evaluation of the integrals in Eqs. \ref{['eq:B_def']} and \ref{['eq:f_def']}. Crosses present the large-$N$ asymptotic form in Eq. \ref{['next']}. Empty colored circles show the intermediate-$N$ asymptotic form in Eq. \ref{['eq:tt0']}, whereas the filled circles indicate the asymptotic relation \ref{['inter']}. Colored vertical thin dash-dotted lines represent $N_\gamma\approx148$, $N_\gamma\approx2.2\times10^4$ and $N_\gamma\approx3.3\times10^6$, respectively.
• Figure 3: (Color online) Coefficient of variation $\kappa$ from Eq. \ref{['kappa']} as function of $N$ for three values of $\gamma$.
• Figure 4: Mean fFPT $\overline{\mathcal{T}_N}$ as a function of $N$ for the dynamics generated by Riemann-Liouville dichotomous noise for three values of the scaling exponent: $\alpha = 0.5$ (subdiffusive behavior, blue), $\alpha = 1$ (diffusive behavior, green) and $\alpha = 1.5$ (superdiffusive behavior, red). The symbols depict the results of simulations with $M=10^4$ particles, for $T_0=1$, $D=1$, $v = 1$, $\lambda=v^2/(2D)$, and $x_0=5$ (such that $x_0/(v T_0) > 1$). The horizontal dashed lines indicate the corresponding ballistic travels times in Eq. \ref{['eq:tmin_H']}.
• Figure S5: CDF of the FPT to the absorbing origin on the halfline, with $v=1$ and $\lambda=0.5$ such that $D=1$, and two values of $x_0$ as shown in the caption. The solid line represents Eq. (\ref{['eq:Stx']}), filled symbols show the empirical CDF from Monte Carlo simulations with $M=10^4$ particles.