Anomalous diffusion in umbrella comb

TL;DR

This paper investigates anomalous diffusion in a circular comb model with two boundary condition scenarios, revealing different diffusion behaviors and stationary distributions, and introduces the umbrella comb as a topologically constrained diffusion model.

Contribution

It introduces the umbrella comb model, analyzing anomalous transport under reflecting and periodic boundary conditions, and links the diffusion behaviors to conformal mappings and Bernoulli polynomials.

Findings

Radial diffusion follows a log-normal distribution with exponential MSD growth.

Periodic boundary conditions lead to normal diffusion in the radial direction.

Stationary distributions are expressed as Bernoulli polynomials.

Abstract

Anomalous transport in a circular comb is considered. The circular motion takes place for a fixed radius, while radii are continuously distributed along the circle. Two scenarios of the anomalous transport, related to the reflecting and periodic angular boundary conditions, are studied. The first scenario with the reflection boundary conditions for the circular diffusion corresponds to the conformal mapping of a 2D comb Fokker-Planck equation on the circular comb. This topologically constraint motion is named umbrella comb model. In this case, the reflecting boundary conditions are imposed on the circular (rotator) motion, while the radial motion corresponds to geometric Brownian motion with vanishing to zero boundary conditions on infinity. The radial diffusion is described by the log-normal distribution, which corresponds to exponentially fast motion with the mean squared displacement (MSD) of the order of $e^t$. The second scenario corresponds to the circular diffusion with periodic boundary conditions and the outward radial diffusion with vanishing to zero boundary conditions at infinity. In this case the radial motion corresponds to normal diffusion. The circular motion in both scenarios is a superposition of cosine functions that results in the stationary Bernoulli polynomials for the probability distributions.