Anomalous diffusion in the strong scattering limit: A Lévy walk approach

Abstract

The continuous time random walk (CTRW) is a powerful stochastic theory developed and used to analyze regular and anomalous diffusion. In particular this framework has been applied to sublinear, dispersive, transport and to enhanced Lévy walks. In its earlier version the CTRW does not include the velocities of the walker explicitly, and therefore it is not suited to analyze situations with randomly distributed velocities. Experiments and theory have recently considered systems which exhibit anomalous diffusion and are characterized by an inherent distribution of velocities. Here we develop a modified CTRW formalism, based on a velocity picture in the strong scattering limit, with emphasis on the Lévy walk limit. We consider a particle which randomly collides with unspecified objects changing randomly its velocity. In the time intervals between collision events the particle moves freely. Two probability density functions (PDF) describe such a process: (a) q(τ), the PDF of times between collision events, and (b) F(v), the PDF of velocities of the particle. In this renewal process both the velocity of the random walker and the time intervals between collision events are independent, identically distributed, random variables. When either q(τ) or F(v) are long-tailed the diffusion may become non-Gaussian. The probability density to find the random walker at r at time t, ρ(r, t), is found in Fourier-Laplace space. We discuss the role of initial conditions especially on the way P(v, t), the probabilty density that the particle has a velocity v at time t, decays to its equilibrium. The phase diagram of the regimes of enhanced, sublinear and normal types of diffusion is presented. We discuss the differences and similarities between the Lévy walk collision process considered here and the CTRW for jump processes.