Pre-asymptotic analysis of Lévy flights

Area of Science:

• Physics
• Statistical Mechanics
• Non-linear Dynamics

Background:

• Lévy flights are anomalous diffusion processes characterized by heavy-tailed random steps.
• Understanding their behavior before reaching the diffusive limit is crucial for various applications.
• Previous studies often focused on tail distributions or the asymptotic diffusive regime.

Purpose of the Study:

• To investigate the bulk properties of Lévy flight distributions at sub-diffusive timescales.
• To identify and characterize timescales beyond the Markovian time-lag in Lévy flights.
• To assess the implications of these findings for real-world applications of Lévy flights.

Main Methods:

• Analysis based on analogs of the Kramers-Moyal expansion.
• Application of the Pawula theorem to characterize the distribution.
• Focus on the bulk of the walker distribution, not its tails.

Main Results:

• For Lévy flights with index α ≤ 2/3, the bulk distribution occurs at wave-numbers greater than (2/α)¹/(2α) ≥ 1.
• The bulk distribution remains non-self-similar for a timescale significantly longer than the Markovian time-lag.
• A distinct, longer timescale, independent of α, governs the transition to the diffusive limit.

Conclusions:

• Lévy flights possess multiple timescales, with a longer one controlling the approach to the diffusive limit.
• The non-self-similar behavior in the bulk distribution challenges assumptions in some applications.
• Reliability of Lévy flight models for optimal searching, foraging, and site fidelity may be compromised by this longer timescale.