# Heterogeneous diffusion in comb and fractal grid structures

## TL;DR

This paper provides exact analytical solutions for diffusion in comb and fractal grid structures with position-dependent diffusion coefficients, revealing various anomalous diffusion regimes depending on structural parameters.

## Contribution

It introduces exact solutions for diffusion behavior in comb and fractal grid structures with power-law dependent diffusion coefficients, highlighting different anomalous regimes.

## Key findings

- Mean square displacement varies as t^{1/(2-α)} for comb structures.
- Fractal grid structures show mean square displacement scaling as t^{(1+ν)/(2-α)}.
- Probability distributions are expressed using Fox H-functions.

## Abstract

We give an exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior $\langle x^2(t)\rangle\sim t^{1/(2-\alpha)}$, where $\alpha$ is the power-law exponent of the position dependent diffusion coefficient $D(x)\sim |x|^{\alpha}$. Depending on the value of $\alpha$ we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., $\langle x^2(t)\rangle\sim t^{(1+\nu)/(2-\alpha)}$, where $0<\nu<1$ is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the Fox $H$-functions.