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Space-fractional Fokker-Planck equation and optimization of random search processes in the presence of an external bias

  • Based on the space-fractional Fokker-Planck equation with a delta-sink term, we study the efficiency of random search processes based on Levy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Levy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Levy flights, due to which Levy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker-Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Levy flights over the regular Brownian search. Contrary to the common belief that Levy flights with a Levy index alpha = 1 (i.e. Cauchy flights) are optimal forBased on the space-fractional Fokker-Planck equation with a delta-sink term, we study the efficiency of random search processes based on Levy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Levy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Levy flights, due to which Levy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker-Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Levy flights over the regular Brownian search. Contrary to the common belief that Levy flights with a Levy index alpha = 1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for alpha may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.show moreshow less

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Metadaten
Author:Vladimir V. Palyulin, Aleksei V. ChechkinORCiDGND, Ralf MetzlerORCiDGND
DOI:https://doi.org/10.1088/1742-5468/2014/11/P11031
ISSN:1742-5468 (print)
Parent Title (English):Journal of statistical mechanics: theory and experiment
Publisher:IOP Publ. Ltd.
Place of publication:Bristol
Document Type:Article
Language:English
Year of first Publication:2014
Year of Completion:2014
Release Date:2017/03/27
Tag:diffusion; driven diffusive systems (theory); fluctuations (theory); stochastic processes (theory)
Pagenumber:32
Funder:Deutsche Forschungsgemeinschaft [PA 2042/1-1]; Academy of Finland within the FiDiPro scheme; DAAD
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Peer Review:Referiert