Strong anomalous diffusion in two-state process with L\'{e}vy walk and Brownian motion

TL;DR

This paper explores strong anomalous diffusion in a two-state process combining Lévy walk and Brownian motion, revealing unique scaling behaviors and distribution characteristics influenced by power-law sojourn times.

Contribution

It provides a detailed analysis of the scaling and distribution properties of a two-state Lévy walk and Brownian motion process, highlighting conditions for strong anomalous diffusion.

Findings

Strong anomalous diffusion occurs even when Lévy walk exponent < 1 if Brownian phase exponent < Lévy walk exponent.

PDF combines stretched Lévy and Gaussian distributions in the central part due to long Brownian sojourns.

Tail PDF remains dominated by Lévy walk's infinite density, showing ballistic scaling.

Abstract

Strong anomalous diffusion phenomena are often observed in complex physical and biological systems, which are characterized by the nonlinear spectrum of exponents $q\nu(q)$ by measuring the absolute $q$-th moment $\langle |x|^q\rangle$. This paper investigates the strong anomalous diffusion behavior of a two-state process with L\'{e}vy walk and Brownian motion, which usually serves as an intermittent search process. The sojourn times in L\'{e}vy walk and Brownian phases are taken as power law distributions with exponents $\alpha_+$ and $\alpha_-$, respectively. Detailed scaling analyses are performed for the coexistence of three kinds of scalings in this system. Different from the pure L\'{e}vy walk, the phenomenon of strong anomalous diffusion can be observed for this two-state process even when the distribution exponent of L\'{e}vy walk phase satisfies $\alpha_+<1$, provided that $\alpha_-<\alpha_+$. When $\alpha_+<2$, the probability density function (PDF) in the central part becomes a combination of stretched L\'{e}vy distribution and Gaussian distribution due to the long sojourn time in Brownian phase, while the PDF in the tail part (in the ballistic scaling) is still dominated by the infinite density of L\'{e}vy walk.