Area coverage of radial Levy flights with periodic boundary conditions

TL;DR

This paper studies the behavior of two-dimensional Levy flights within a finite area with periodic boundaries, analyzing their coverage, probability distribution, and search efficiency over time.

Contribution

It provides new insights into the ergodic properties and area coverage dynamics of Levy flights with periodic boundary conditions through simulation.

Findings

Fractal path dimension reaches saturation at 2, indicating complete area coverage.

Probability density functions evolve over time and reach equilibrium.

Mean first passage time depends on the stable index of Levy flights.

Abstract

We consider the time evolution of two-dimensional Levy flights in a finite area with periodic boundary conditions. From simulations we show that the fractal path dimension d_f and thus the degree of area coverage grows in time until it reaches the saturation value d_f=2 at sufficiently long times. We also investigate the time evolution of the probability density function and associated moments in these boundary conditions. Finally we consider the mean first passage time as function of the stable index. Our findings are of interest to assess the ergodic behavior of Levy flights, to estimate their efficiency as stochastic search mechanisms and to discriminate them from other types of search processes.