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.
Highlights
In the recent decades, it is widely recognized that anomalous diffusion is a very general phenomenon in the natural world, which is characterized by the nonlinear evolution of mean squared displacement with respect to time, i.e., x2(t) ∝ tβ with β = 1 [1,2,3]
The aim of this paper is to investigate the complementary probability density function (PDF) and the strong anomalous diffusion behavior of this two-state process
Comparing with the thoroughly investigated strong anomalous diffusion behavior of standard Levy walk, a larger exponent α− in our two-state process makes no difference on the diffusion behavior
Summary
The two-state process could exhibit the strong anomalous diffusion even for α < 1 if the sojourn time in Brownian phase is longer than the one of Levy walk phase. We assume that the sojourn time distributions of the two-state process switching between Levy walk and Brownian phase are ψ+(t) and ψ−(t), respectively. The survival probability of finding the sojourn time in state ‘±’ exceeding t is defined as It is well-known that the dynamical behaviors of standard Levy walk vary significantly for different regimes of power law exponents, which naturally motivates us to study the properties of this two-state process with different values of α± ∈ (0, 2)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
