The article presents a counterexample to the widely held belief that random search algorithms using Lévy flights can find a target faster than using Brownian motion. The authors develop three different approaches - Monte Carlo simulations, numerical solutions of (pseudo)-differential equations, and asymptotic analysis - to show that in the narrow capture framework, where a random search is performed for a small stationary target within a confined search domain, Brownian search is more efficient on average than Lévy flight search.
The key insights are:
The global mean first passage time (GMFPT) of the Lévy search increases as the Lévy flight tail index α deviates further from the Brownian limit of 1. This is in contrast to the Brownian search, whose GMFPT scales logarithmically with the target size.
The authors provide a possible explanation for the longer average duration of Lévy searches. Lévy searches have a greater likelihood of taking long jumps away from the target, effectively restarting the search from a farther location. This leads to anomalously long search times compared to Brownian searches.
The asymptotic analysis reveals that the leading order term of the Lévy search GMFPT depends on the geometry of the target, unlike the Brownian case where target geometry effects enter only at higher order. The global geometry of the search domain is encoded in the O(1) correction term.
Overall, the article challenges the prevailing Lévy flight foraging hypothesis and provides a counterexample where Brownian search outperforms Lévy flight strategies.
翻譯成其他語言
從原文內容
arxiv.org
深入探究