Dynamical approach to anomalous diffusion: Response of Lévy processes to a perturbation

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Abstract
Lévy statistics are derived from a dynamical system, which can be either Hamiltonian or not, using a master equation approach. We compare these predictions to the random walk approach recently developed by Zumofen and Klafter for both the nonstationary [Phys. Rev. E 47, 851 (1993)] and stationary [Physica A 196, 102 (1993)] case. We study the unperturbed dynamics of the system analytically and numerically and evaluate the time evolution of the second moment of the probability distribution. We also study the response of the dynamical system undergoing anomalous diffusion to an external perturbation and show that if the slow regression to equilibrium of the variable ‘‘velocity’’ is triggered by the perturbation, the process of diffusion of the ‘‘space’’ variable takes place under nonstationary conditions and a conductivity steadily increasing with time is generated in the early part of the response process. In the regime of extremely long times the conductivity becomes constant with a value, though, that does not correspond to the prescriptions of the ordinary Green-Kubo treatments.

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