Destruction of ultra-slow diffusion in a three dimensional cylindrical comb structure

TL;DR

This paper rigorously analyzes anomalous transport in a 3D comb structure, showing how ultra-slow diffusion can be destroyed or transformed into subdiffusion depending on boundary conditions and diffusion types in secondary branches.

Contribution

It provides a mathematical solution to the Fokker-Planck equation for 3D combs, revealing conditions under which ultra-slow diffusion is destroyed or altered.

Findings

Ultra-slow diffusion occurs with MSD ~ ln(t) in infinite secondary branches.

Finite boundaries lead to a transition from ultra-slow to normal diffusion.

Anomalous diffusion in secondary branches results in enhanced subdiffusion with MSD ~ t^{1-α} ln t.

Abstract

We present a rigorous 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 threaten on a backbone. It is shown that the ultra-slow contaminant 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 $\sim t^{1-\alpha}\ln t$, where $0<\alpha<1$.