Fastest first-passage time for multiple searchers with finite speed

TL;DR

This paper analyzes the mean fastest first-passage time for multiple finite-speed searchers, revealing a lower bound related to ballistic travel time and demonstrating efficiency advantages over Brownian searchers, with extensions to anomalous diffusion.

Contribution

It provides an analytical and numerical study of the fastest first-passage time for finite-speed searchers, highlighting a fundamental lower bound and efficiency benefits over diffusive models.

Findings

Mean fFPT is bounded by minimal ballistic travel time.

Finite-speed searchers outperform Brownian ones in short-time detection.

Superdiffusive regimes enhance target detection efficiency.

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.