Different effects of external force fields on aging L\'{e}vy walk

TL;DR

This paper investigates how external force fields influence aging phenomena in L\'evy walks, revealing that constant forces induce aging and enhance diffusion, whereas periodic forces do not, through theoretical and numerical analysis.

Contribution

It provides a comparative analysis of the effects of constant and periodic external forces on aging L\'evy walks, highlighting the role of constant force in inducing aging phenomena.

Findings

Constant force induces aging and enhances diffusion in L\'evy walks.

Periodic force does not induce aging phenomena.

Theoretical analysis and simulations confirm the different effects of the two forces.

Abstract

Aging phenomena have been observed in numerous physical systems. Many statistical quantities depend on the aging time $t_a$ for aging anomalous diffusion processes. This paper pays more attention to how an external force field affects the aging L\'{e}vy walk. Based on the Langevin picture of L\'{e}vy walk and generalized Green-Kubo formula, we investigate the quantities which include the ensemble- and time-averaged mean-squared displacements in both weak aging $t_a\ll t$ and strong aging $t_a\gg t$ cases, and compare them to the quantities in the absence of any force field. Two typical force fields, constant force $F$ and time-dependent periodic force $F(t)=f_0\sin(\omega t)$, are considered for comparison. The generalized Einstein relation is also discussed in the case with constant force. We find that the constant force is the key of generating the aging phenomena and enhancing the diffusion behavior of aging L\'{e}vy walk, while the time-dependent periodic force is not. The different effects of the two kinds of forces on the aging phenomena of L\'{e}vy walk are verified by both theoretical analyses and numerical simulations.