Computer Simulation Study of the Levy Flight Process

TL;DR

This study uses computer simulations to analyze Levy flight processes, confirming linear mean square displacement over time, exploring effects of jump length restrictions, and suggesting non-ergodic behavior in Levy distributions.

Contribution

It provides a detailed simulation-based analysis of Levy flights, examining how constraints and averaging methods influence diffusion properties and ergodicity.

Findings

Mean square displacement is linearly related to time in Levy flights.

Maximum jump length does not affect linearity but influences diffusion coefficient.

Time averaging reveals non-ergodic characteristics of Levy distributions.

Abstract

Random walk simulation of the Levy flight shows a linear relation between the mean square displacement <r2> and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value (lm) affects the diffusion coefficient, even though it remains constant for lm greater than 1464. So, this factor has no effect on the linearity. In addition, it is shown that the number of samples does not affect the results. We have demonstrated that the relation between the mean square displacement and time remains linear in a continuous space, while continuous variables just reduce the diffusion coefficient. The results are also implied that the movement of a levy flight particle is similar to the case the particle moves in each time step with an average length of jumping <l>. Finally, it is shown that the non-linear relation of the Levy flight will be satisfied if we use time average instead of ensemble average. The difference between time average and ensemble average results points that the Levy distribution may be a non-ergodic distribution.