TY - JOUR

T1 - Space-fractional Fokker-Planck equation and optimization of random search processes in the presence of an external bias

AU - Palyulin, Vladimir V.

AU - Chechkin, Aleksei V.

AU - Metzler, Ralf

PY - 2014/11/1

Y1 - 2014/11/1

N2 - Based on the space-fractional Fokker-Planck equation with a δ-sink term, we study the efficiency of random search processes based on Lévy 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 Lévy 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 Lévy flights, due to which Lévy 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 Lévy flights over the regular Brownian search. Contrary to the common belief that Lévy flights with a Lévy index α=1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for α may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.

AB - Based on the space-fractional Fokker-Planck equation with a δ-sink term, we study the efficiency of random search processes based on Lévy 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 Lévy 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 Lévy flights, due to which Lévy 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 Lévy flights over the regular Brownian search. Contrary to the common belief that Lévy flights with a Lévy index α=1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for α may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.

KW - diffusion

KW - driven diffusive systems (theory)

KW - fluctuations (theory)

KW - stochastic processes (theory)

UR - http://www.scopus.com/inward/record.url?scp=84914706610&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/2014/11/P11031

DO - 10.1088/1742-5468/2014/11/P11031

M3 - Article

AN - SCOPUS:84914706610

VL - 2014

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 11

M1 - P11031

ER -