Levy walks with velocity fluctuations

TL;DR

This paper extends the standard Levy walk model by incorporating velocity fluctuations, providing analytical solutions for different models and highlighting how initial conditions influence diffusion profiles, supported by numerical simulations.

Contribution

It introduces and analytically evaluates two new models of Levy walks with velocity fluctuations, advancing understanding of their diffusion behavior.

Findings

Velocity fluctuations significantly affect diffusion profiles.

Initial conditions influence the ballistic regions of the process.

Analytical results are validated by numerical simulations.

Abstract

The standard Levy walk is performed by a particle that moves ballistically between randomly occurring collisions, when the intercollision time is a random variable governed by a power-law distribution. During instantaneous collision events the particle randomly changes the direction of motion but maintains the same constant speed. We generalize the standard model to incorporate velocity fluctuations into the process. Two types of models are considered, namely, (i) with a walker changing the direction and absolute value of its velocity during collisions only, and (ii) with a walker whose velocity continuously fluctuates. We present full analytic evaluation of both models and emphasize the importance of initial conditions. We show that the type of the underlying Levy walk process can be identified by looking at the ballistic regions of the diffusion profiles. Our analytical results are corroborated by numerical simulations.