Exact Results for the Nonergodicity of Generalized Lévy Walks

We study the generalized Lévy walk introduced by Shlesinger et. al. [1], which was introduced to explain Richardson's t³-law for turbulent diffusion of passive scalars. This model, which can describe subdiffusive, diffusive, or superdiffusive motion, is considered as very physical because of the continuous nature of its trajectories. It applies to many processes in physics and biology, where velocity and duration or distance of travel are coupled. In extension of our results for integrated Brownian motion [2], we are able to obtain exact results for the ensemble- and the time-averaged squared displacement, and for the ergodicity breaking parameter in the full parameter space of this model. In certain regions of the latter we obtain surprising results such as the divergence of the mean-squared displacements, at variance with the t³-law, the divergence of the ergodicity breaking parameter despite a finite mean-squared displacement, and also subdiffusion which appears superdiffusive when one only considers time averages.
[1] M. Shlesinger, B. West and J. Klafter, Phys. Rev. Lett. 58, 1100 (1987)
[2] T. Albers and G. Radons, Phys. Rev. Lett. 113, 184101 (2014)