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.
In many biological contexts, diverse agents must efficiently locate distant targets-such as transcription factors binding to specific operators on the cellular DNA, immune cells detecting pathogens, or signaling molecules reaching their conjugate receptors [1][2][3]. These processes are inherently stochastic and often proceed in crowded dynamic environments such as cellular cytoplasm. Although the first-passage statistics for individual searchers are quite well understood [4][5][6], biological systems rarely rely on a single agent; instead, multiple searchers are deployed in parallel to speed up target detection [2,7,8].
When N searchers begin simultaneously, target detection is determined not by a single agent’s shortest arrival time τ , but competitively by the earliest arrival time among all searchers. The problem thus shifts from individual first-passage times τ k , k = 1, 2, . . . , N , to the statistic of the extremal random variable T N = min{τ 1 , . . . , τ N }-the fastest first-passage time (fFPT). Order statistics therefore play a central role: the key aspects are the fFPT distribution and how its moments scale with N , characterizing the efficiency of such a multiagent search. The analysis of such multiple searcher dynamics reveals how redundancy improves both the reliability and speed of biological search, offering insight into how living systems may optimize their performance.
Most existing analyses of multi-agent search assume that all agents start from the same point simultaneously and move independently as Brownian walkers with the same diffusion coefficient D [9][10][11][12][13][14][15][16]. Their positions follow Langevin dynamics with Gaussian white noise, and the position probability density function (PDF) of each searcher evolves according to the diffusion equation from t = 0, a framework that accurately captures the longtime behavior of many stochastic processes in nature. Within this setting, the mean fFPT to a target at a dis-tance x 0 follows the inverse-logarithmic law [9][10][11][12][13][14][15][16]
Here the bar denotes averaging with respect to individual trajectory realizations of all searchers, and the symbol “≃” signifies that we consider solely the leading-order behavior in the limit N → ∞. This result was first derived for one-dimensional continuous-space systems, but later shown to hold in bounded domains of any dimension [17,18], because it is dominated by so-called “direct” trajectories [19,20] that go straight to the target, rendering the actual embedding spatial dimension irrelevant. The asymptotic form (1) shows that deploying more and more searchers steadily lowers the mean fFPT, albeit only logarithmically with N . In the limit N → ∞, the fastest searcher would thus reach the distant target arbitrarily quickly-effectively instantaneously. Moreover, assuming that the agents undergo a subdiffusive motion with the mean-squared displacement (MSD) x 2 (t) ∝ t α (0 < α < 1) and a position PDF obeying a fractional diffusion equation [21], it was shown that
where t α is a characteristic time-scale [22]. This asymptotic result indicates that for subdiffusive dynamics, T N also vanishes as N → ∞. Strikingly, Eq. ( 2) implies that T N → 0 when α → 0, suggesting that slower diffusion enhances the speed of the fastest arrival, as highlighted in the title of [22]. Although mathematically rigorous, the results (1) and ( 2) are clearly unrealistic-even the fastest searcher needs a finite time to reach a target a finite distance away-thus underscoring the need for a more refined analysis that yields more plausible behavior.
Here, we revisit this long-standing problem by assuming that individual searchers follow a one-dimensional generalized Langevin dynamics driven by symmetric dichotomous noise-a stochastic motion with random switching between velocities ±v with rate λ [23,24] (see also [25,26] and references therein). This choice avoids the unphysical behavior inherent in Gaussian white-noise models, where a searcher has a non-zero probability of appearing arbitrarily far from its starting point in arbitrarily short time, allowing for unrealistically small first-passage times (historically, this problem is well known in heat transport, where it is circumvented by replacing the parabolic diffusion equation by a hyperbolic Cattaneo equation with finite propagation speed [27,28]). For large N , such short-time artifacts dominate the moments of T N , over-estimating the survival probabilities. Dichotomous noise thus provides a more realistic framework to capture the shorttime dynamics relevant to multi-agent search. Moreover, symmetric dichotomous noise with alternating velocities naturally encodes strong antipersistence-common in crowded environments-since each forward step is followed by a backward one (see, e.g., [29]). We also note that this very framework has been successfully used to model bacterial and other active-particle dynamics, yielding physically realistic behavior (
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