Superdiffusion in Scale-Free Inhomogeneous Environments

We investigate the impact of external potentials on superdiffusive random walks known as Levy flights and show that even strongly superdiffusive transport is substantially affected by external fields. Unlike ordinary random walks, Levy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Levy index. Our analysis is based on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium.Thus, the results presented are applicable to the large class of situations in which superdiffusion is caused by topological complexity, such as diffusion on folded polymers and scale-free networks. We investigate periodic potentials as well as random potentials and compare the novel approach to generalized Langevin dynamics. Furthermore, we introduce a model for the generation of human saccadic eye-movements which can be mapped onto the generalized Fokker-Planck equation and show that eye-movements can be understood in terms of a temporal optimization scheme.