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.

• The paper derives an exact mean fastest first-passage time for N≥3 dichotomous-noise searchers, proving it is bounded below by the ballistic travel time x₀/v.
• Finite-speed dynamics produces exponential convergence to the ballistic limit with increasing N, contrasting with Brownian models’ inverse-logarithmic law and preventing unphysical instantaneous arrivals.
• The paper shows that diffusion and infinite-searcher limits do not commute and that finite-speed anomalous transport generally ranks superdiffusion faster than normal diffusion and subdiffusion.

The mean fastest first-passage time (fFPT) among $N$ independent searchers is a central quantity in the theory of parallel target search. For Brownian searchers released simultaneously from a common point, the mean fFPT obeys the inverse-logarithmic law $\overline{T_N}\simeq x_0^2/(4D\ln N)$ as $N\to\infty$, implying that the fastest searcher reaches a target at distance $x_0$ arbitrarily quickly. Grebenkov, Metzler, and Oshanin revisit this problem by replacing the Gaussian white-noise Langevin dynamics with finite-speed dichotomous noise, and show that the unphysical short-time behavior of diffusive models is responsible for these artifacts. In the finite-speed setting, the mean fFPT is bounded below by the ballistic travel time $t_{\min}=x_0/v$ and converges to this bound exponentially fast in $N$ (2602.15627).

Physical motivation and model

The authors consider $N$ particles released at $x_0>0$ at time $t=0$, searching for an immobile absorbing target at the origin on the positive half-line. Each particle obeys $\dot{x}_k(t)=\eta_k(t)$, where $\overline{T_N}\simeq x_0^2/(4D\ln N)$0 is an independent symmetric dichotomous noise alternating between velocities $\overline{T_N}\simeq x_0^2/(4D\ln N)$1 with switching rate $\overline{T_N}\simeq x_0^2/(4D\ln N)$2, with long-time diffusion coefficient $\overline{T_N}\simeq x_0^2/(4D\ln N)$3. The single-particle survival probability satisfies the telegrapher's equation, so the position PDF has compact, time-dependent support and no probability leaks beyond the ballistic light cone. The key dimensionless control parameter is $\overline{T_N}\simeq x_0^2/(4D\ln N)$4, the ballistic travel time measured in units of the noise correlation time. The authors argue that $\overline{T_N}\simeq x_0^2/(4D\ln N)$5 spans physically relevant values: it can be very large ($\overline{T_N}\simeq x_0^2/(4D\ln N)$6--$\overline{T_N}\simeq x_0^2/(4D\ln N)$7 per micrometer) for proteins or ions in aqueous solution, moderate ($\overline{T_N}\simeq x_0^2/(4D\ln N)$8--$\overline{T_N}\simeq x_0^2/(4D\ln N)$9 per micrometer) in crowded cytoplasm, and of order unity for E. coli run-and-tumble motion, for which they estimate $N\to\infty$0m using experimentally determined parameters.

This parameter estimate matters because it determines which asymptotic regime of $N\to\infty$1 is actually observable in a given physical setting. The dichotomous model is also directly relevant to active-particle dynamics, including sperm-cell motility that motivated earlier fFPT studies.

Exact result and the ballistic bound

The central analytical result is exact for all $N\to\infty$2 and $N\to\infty$3:

$N\to\infty$4

where the dimensionless excess factor $N\to\infty$5 is expressed through modified Bessel functions. Since $N\to\infty$6, the ballistic travel time $N\to\infty$7 is a rigorous lower bound on the mean fFPT—this is the qualitative departure from the Brownian theory, in which no such bound exists. The excess factor is monotonically increasing in $N\to\infty$8 and decreasing in $N\to\infty$9, and the moments for $x_0$0 diverge, as in ordinary diffusion on the half-line, so the analysis is restricted to $x_0$1.

The derivation builds on the exact single-particle survival probability obtained from the backward Fokker–Planck equations with switching. The survival probability is strictly unity for $x_0$2, then jumps by $x_0$3—the probability of reaching the target in a single ballistic run—before decaying with the Lévy–Smirnov $x_0$4 tail at long times. The authors verify this macroscopic description against Monte Carlo simulations and find excellent agreement even for moderate $x_0$5, where only a few velocity switches are involved.

Exponential convergence at large $x_0$6

For fixed $x_0$7 and $x_0$8, where

$x_0$9

the excess factor decays exponentially:

$t_{\min}=x_0/v$0

The convergence of $t_{\min}=x_0/v$1 to the ballistic bound is therefore exponential in $t_{\min}=x_0/v$2, in stark contrast to the logarithmic improvement predicted by Brownian theory. The coefficient of variation $t_{\min}=x_0/v$3 of the fFPT decays exponentially as well, $t_{\min}=x_0/v$4, so for large $t_{\min}=x_0/v$5 the mean fFPT is a faithful, self-averaging characterization of the search dynamics. Biologically, this means that investing in many parallel searchers yields far greater speedup than the Brownian redundancy principle suggests.

Intermediate regime and non-commuting limits

For large $t_{\min}=x_0/v$6, the threshold $t_{\min}=x_0/v$7 grows exponentially (e.g., $t_{\min}=x_0/v$8 at $t_{\min}=x_0/v$9 but exceeds $N$0 at $N$1), so the exponential regime is unobservable for realistic $N$2 in aqueous systems. The physically relevant regime is then $N$3, where the authors derive, via a large-argument Bessel approximation and Williams' tight bound on the error function, the uniform asymptotic form

$N$4

with $N$5 and $N$6 the harmonic number. In the double limit $N$7 followed by $N$8, this yields $N$9, recovering the inverse-logarithmic law up to the numerical prefactor $N$0 instead of $N$1. Crucially, taking the diffusion limit first ($N$2 at fixed $N$3, so $N$4) and then $N$5 reproduces the Brownian result, whereas taking $N$6 first yields the ballistic bound. The two limits do not commute, and the authors identify the logarithmic Brownian law as an artifact of the ordering in which the singular diffusion limit is taken. This reconciles earlier results with finite-speed dynamics: for intermediate $N$7 and large $N$8, the dichotomous model mimics the familiar logarithmic behavior.

Anomalous diffusion

The authors extend the analysis to anomalous transport using the Riemann–Liouville fractional dichotomous process, $N$9 with power-law kernel $x_0>0$0, $x_0>0$1, which preserves finite-speed propagation inside a growing "light cone" $x_0>0$2. For $x_0>0$3, the minimal travel time is

$x_0>0$4

a decreasing function of $x_0>0$5. Simulations with $x_0>0$6 particles confirm the hierarchy: superdiffusive search is fastest, followed by normal diffusion, then subdiffusion—both in the minimal time and in the rate of convergence with $x_0>0$7. This directly contradicts the prediction from continuous-space fractional diffusion equations that $x_0>0$8 decreases with decreasing $x_0>0$9 (i.e., that subdiffusion is "faster" than normal diffusion for the fastest searcher). The authors note that the counter-intuitive trend reappears for $t=0$0, where the target can be reached in a single move, but argue that this regime is artificial since first passage is governed by short times, where larger-$t=0$1 processes actually spread more slowly.

Limitations and open questions

Several caveats are explicit in the paper. The exact mean fFPT is restricted to $t=0$2; the variance in the intermediate regime $t=0$3 is not analyzed asymptotically but only computed numerically. The intermediate-$t=0$4 asymptotics assume $t=0$5 bounded away from zero and rely on an error-function approximation of $t=0$6 whose accuracy at the point $t=0$7 is only formal (exact only as $t=0$8), though numerical checks show excellent agreement over broad ranges. The anomalous-diffusion results are obtained by simulation rather than closed-form analysis, and the hierarchy reversal for $t=0$9 shows that the superdiffusion-beats-subdiffusion conclusion is parameter-dependent. The paper also leaves open the determination of the full fFPT distribution beyond the mean and variance, and the extension to discrete-space (network) searchers.

Conclusion

By imposing finite propagation speed through dichotomous noise, this work shows that the extreme first-passage statistics of parallel search are qualitatively different from Brownian predictions: the mean fFPT is bounded by the ballistic travel time and approaches it exponentially fast in the number of searchers, while the familiar inverse-logarithmic law emerges only in an intermediate regime or from a specific, non-commuting order of limits. The framework also restores the physically expected hierarchy in which superdiffusion outperforms normal diffusion and subdiffusion. The results indicate that diffusive macroscopic models can be misleading for short-time, extreme-value behavior, and that finite-velocity search dynamics is the appropriate description for rapid parallel target detection in crowded biological environments.