Strong anomalous diffusion in two-state process with Lévy walk and Brownian motion

Abstract

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