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 $\alpha$.

arXiv comment11 pages, 2 figures

References in corpus (26)

• Lévy walks
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking
• Langevin equations for continuous time Lévy flights
• Diffusion on a solid surface: Anomalous is normal
• Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement
• Theory of fractional-Lévy kinetics for cold atoms diffusing in optical lattices
• Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem
• Finite-time effects and ultraweak ergodicity breaking in superdiffusive dynamics
• Joint Probability Distributions for a Class of Non-Markovian Processes
• Time averaged Einstein relation and fluctuating diffusivities for the Lévy walk
• Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion
• Subordinated Langevin Equations for Anomalous Diffusion in External Potentials - Biasing and Decoupled Forces
• Scaling Green-Kubo relation and application to three aging systems
• Anomalous processes with general waiting times: functionals and multi-point structure
• Enhancement, slow relaxation, ergodicity and rejuvenation of diffusion in biased continuous-time random walks
• Space-time velocity correlation function for random walks
• Lévy-walk-like Langevin dynamics
• Subdiffusion in an external force field
• Sub-diffusion in External Potential: Anomalous hiding behind Normal
• Tempered fractional Langevin-Brownian motion with inverse $\beta$-stable subordinator
• Scale invariant Green-Kubo relation for time averaged diffusivity
• Feynman-Kac equation for anomalous processes with space- and time-dependent forces
• A no-go theorem for ergodicity and an Einstein relation
• Langevin formulation of a subdiffusive continuous time random walk in physical time
• Aging two-state process with Lévy walk and Brownian motion

Cited by in corpus (6)

• 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
• Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential
• Lévy walk dynamics in mixed potentials from the perspective of random walk theory
• Random diffusivity processes in an external force field