# Comb Model in Periodic Potential

## TL;DR

This paper introduces a comb model with periodic potential in side branches to study how it affects transport and leads to non-equilibrium stationary states.

## Contribution

The paper introduces a generalized comb model with periodic potential in side branches and derives exact results for its transport behavior.

## Key findings

- A non-equilibrium stationary state (NESS) occurs in the comb geometry when the total energy is zero.
- The probability density near NESS follows a Mathieu distribution with zero energy.
- The model can describe anisotropic particle dispersion in atmospheric or plasma turbulence and the formation of layered structures.

## Abstract

A comb model with periodic potential in side branches is introduced. A comb model is a model of geometrically constrained diffusion, such that the diffusion process along the comb’s main axis (backbone) is coupled to the diffusion process in fingers, the side branches of the comb. Here, we consider a generalized version of this complex process by enabling a periodic potential function in the fingers. We aim to understand how the potential function added affects the asymptotic transport scalings in the backbone. A set of exact results pertaining to the generalized model is obtained. It is shown that the relaxation process in fingers leads directly to the occurrence of a non-equilibrium stationary state (NESS) in comb geometry, provided that the total energy is zero. Also, it is shown that the spatial distribution of the probability density in proximity to NESS is given by the Mathieu distribution with zero energy. The latter distribution is found to be the direct result of relaxation towards stationarity of the Mathieu eigenspectrum. It is suggested that the generalized model can characterize anisotropic particle dispersion in beta-plane atmospheric (alternatively, electrostatic drift-wave plasma) turbulence and the subsequent formation of layered structures, zonal flows, and staircases. In this regard, the inherent interconnection between combs and staircases is discussed in some detail.