Does ultra-slow diffusion survive in a three dimensional cylindrical comb?

Abstract

We present an exact analytical result on ultra-slow diffusion by solving a Fokker–Planck equation, which describes anomalous transport in a three dimensional (3D) comb. This 3D cylindrical comb consists of a cylinder of discs of either infinite or finite radius, threaded on a backbone. It is shown that the ultra-slow particle spreading along the backbone is described by the mean squared displacement (MSD) of the order of ln (t). This phenomenon takes place only for normal two dimensional diffusion inside the infinite secondary branches (discs). When the secondary branches have finite boundaries, the ultra-slow motion is a transient process and the asymptotic behavior is normal diffusion. In another example, when anomalous diffusion takes place in the secondary branches, a destruction of ultra-slow (logarithmic) diffusion takes place as well. As the result, one observes “enhanced” subdiffusion with the MSD ∼t1−αln(t), where 0 < α < 1.