Langevin picture of Lévy walk in a constant force field

Abstract

Lévy 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 Lévy 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 Lévy 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 $\alpha$.

11 pages, 2 figures

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• Lévy walks
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Space-time velocity correlation function for random walks
• Subdiffusion in an external force field
• Sub-diffusion in External Potential: Anomalous hiding behind Normal
• Scale invariant Green-Kubo relation for time averaged diffusivity
• Langevin formulation of a subdiffusive continuous time random walk in physical time

Cited by in corpus (5)

• Ergodic property of Langevin systems with superstatistical, uncorrelated or correlated diffusivity
• Lévy-walk-like Langevin dynamics affected by a time-dependent force
• Langevin picture of anomalous diffusion processes in expanding medium
• Lévy walk dynamics in mixed potentials from the perspective of random walk theory
• Random diffusivity processes in an external force field