Lévy walks with variable waiting time: A ballistic case.

Area of Science:

• Physics
• Statistical Mechanics
• Complex Systems

Background:

• Lévy walk processes describe particle movement with non-exponential waiting times.
• Understanding position-dependent waiting times is crucial for modeling complex transport phenomena.

Purpose of the Study:

• To analyze the Lévy walk process with position-dependent waiting times for fractional anomalous diffusion.
• To investigate the impact of rising and diminishing waiting time rates on particle density distributions.

Main Methods:

• Derivation of density distributions using fractional equations for resting and in-flight particles.
• Approximation of the master equation for rising and diminishing waiting time rates.
• Comparison of analytical results with Monte Carlo trajectory simulations.

Main Results:

• For diminishing waiting time rates, particle density follows an alpha-stable distribution, leading to Lévy flights with enhanced, non-ballistic diffusion.
• Rising waiting time rates result in different diffusion characteristics.
• Analytical findings qualitatively align with observed human and animal movement patterns.

Conclusions:

• Position-dependent waiting times significantly alter Lévy walk dynamics, leading to distinct diffusion regimes.
• The model provides insights into the mechanisms underlying complex biological movements.
• Fractional equations offer a powerful framework for analyzing anomalous diffusion processes.