Scale Dependent Dimension in the Forest Fire Model

TL;DR

This paper investigates the scale-dependent fractal dimension of the forest fire model, revealing a continuous variation with scale and potential applications to turbulence and cosmic matter distribution.

Contribution

It introduces the concept of a scale-dependent fractal dimension in the forest fire model, linking it to turbulence and cosmic structures.

Findings

Fractal dimension varies from 0 to 3 in 3D model with scale

Correlation length diverges as p^{-2/3}

Dimension jumps from 1 to 2 at specific scale in 2D model

Abstract

The forest fire model is a reaction-diffusion model where energy, in the form of trees, is injected uniformly, and burned (dissipated) locally. We show that the spatial distribution of fires forms a novel geometric structure where the fractal dimension varies continuously with the length scale. In the three dimensional model, the dimensions varies from zero to three, proportional with $log(l)$, as the length scale increases from $l \sim 1$ to a correlation length $l=\xi$. Beyond the correlation length, which diverges with the growth rate $p$ as ${\xi} \propto p^{-2/3}$, the distribution becomes homogeneous. We suggest that this picture applies to the ``intermediate range'' of turbulence where it provides a natural interpretation of the extended scaling that has been observed at small length scales. Unexpectedly, it might also be applicable to the spatial distribution of luminous matter in the universe. In the two-dimensional version, the dimension increases to D=1 at a length scale $l \sim 1/p$, where there is a cross-over to homogeneity, i. e. a jump from D=1 to D=2.