A dynamical approach to anomalous conductivity

Abstract

We study a process of anomalous diffusion of a variable resulting from the fluctuations of a dichotomous velocity whose two states, in the absence of perturbation, have the same waiting time distribution ψ(t). In the long-time limit the function ψ(t) is proportional tot ^−μ with 2<μ<3. Previously this distribution along with the constraint on μ proved to be a dynamical realization of an α-stable Lévy process with α=μ−1. Here we study the response of this anomalous diffusion process to a perturbation which has the effect of truncating the inverse power law of one of the two states of the velocity for timest>1/ε, where ε is proportional to the intensity of the weak perturbation. We show that the resulting transport process is characterized by a succession of two regimes: the first still satisfies the prescriptions of the Green-Kubo approach to conductivity, and, in accordance with the nature of the anomalous diffusion studied here, corresponds to a state of increasing conductivity (IC); the second regime is characterized by a constant conductivity (CC). The transition from the IC to the CC regime takes place in a time of the order oft∼1/ε and consequently the transition occurs at longer and longer times, as the perturbation intensity decreases. The final stationary regime corresponds to an asymmetric Lévy process of diffusion.