Subdiffusion in an array of solid obstacles

Abstract

More than a decade ago, Goychuk reported on a universal behavior of subdiffusive motion (as described by the generalized Langevin equation) in a one-dimensional bounded periodic potential (Goychuk 2009 Phys. Rev. E 80 046125) where the numerical findings show that the long-time behavior of the mean squared displacement is not influenced by the potential, so that the behavior in the potential, under homogenization, is the same as in its absence. This property may break down if the potential is unbounded. In the present work, we report on the results of simulations of subdiffusion in a two-dimensional (2D) periodic array of solid obstacles (i.e. in an unbounded potential) with different packing fractions. It is revealed that the universal subdiffusive behavior at long times is not influenced by the presence of solid scatterers, whose presence influences the behavior at intermediate times only. This result is discussed as having possible relations to the emerging problem of interpretation of results on trajectories of tracers spreading in the brain’s extracellular space.