Langevin picture of L\'{e}vy walk in a constant force field

TL;DR

This paper investigates how a constant external force affects Lévy walks, revealing faster diffusion, super-ballistic behavior, and non-ergodicity through a subordinated Langevin framework and simulations.

Contribution

It introduces a subordinated Langevin model incorporating a constant force to analyze Lévy walks, highlighting its effects on diffusion and ergodic properties.

Findings

Constant force accelerates particle diffusion.

Super-ballistic diffusion observed under force.

Non-ergodic behavior confirmed by simulations.

Abstract

L\'{e}vy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the L\'{e}vy walk with the exponent of the power-law distributed flight time $\alpha\in(0,2)$. We add the term $F\eta(s)$ ($\eta(s)$ is the L\'{e}vy noise) on a subordinated Langevin system to characterize such a constant force, being effective on the velocity process for all physical time after the subordination. We clearly show the effect of the constant force $F$ on this Langevin system and find this system is like the continuous limit of the collision model. The first moments of velocity processes for these two models are consistent. In particular, based on the velocity correlation function derived from our subordinated Langevin equation, we investigate more interesting statistical quantities, such as the ensemble- and time-averaged mean squared displacements. Under the influence of constant force, the diffusion of particles becomes faster. Finally, the super-ballistic diffusion and the non-ergodic behavior are verified by the simulations with different $\alpha$.