L\'evy walks in nonhomogeneous environments

TL;DR

This paper investigates Le9vy walks with rests in nonhomogeneous environments, deriving a master equation, analyzing diffusion behavior, and demonstrating how trap heterogeneity can lead to subdiffusion instead of enhanced diffusion.

Contribution

It introduces a model of Le9vy walks with position-dependent trapping, deriving the master equation and analyzing the impact of heterogeneity on diffusion properties.

Findings

Asymptotic distributions follow a stretched-exponential shape.

Heterogeneous traps can cause a transition from superdiffusion to subdiffusion.

Monte Carlo simulations confirm the theoretical predictions.

Abstract

The L\'evy walk process with rests is discussed. The jumping time is governed by an $\alpha$-stable distribution with $\alpha>1$ while a waiting time distribution is Poissonian and involves a position-dependent rate which reflects a nonhomogeneous trap distribution. The master equation is derived and solved in the asymptotic limit for a power-law form of the jumping rate. The relative density of resting and flying particles appears time-dependent and the asymptotic form of both distribution obey a stretched-exponential shape at large time. The diffusion properties are discussed and it is demonstrated that, due to the heterogeneous trap structure, the enhanced diffusion, observed for the homogeneous case, may turn to a subdiffusion. The density distributions and mean squared displacements are also evaluated from Monte Carlo simulations of individual trajectories.