The fractal nature of a diffusion front and the relation to percolation

Abstract

International audienceUsing a two dimensional simulation, a diffusion front is shown to have a fractal geometry in a range increasing with the diffusion length. The number of particles on the front, and the width measuring its spread, follow power laws as a function of the diffusion length. The associated exponents and the fractal dimension can be expressed as simple functions of the critical exponents of the two dimensional percolation problem

Modeling urban growth patterns with correlated percolation

“…Our "physical" model [10], which corresponds to the correlated percolation model [16][17][18][19][20] in the presence of a density gradient [21][22][23], is motivated by the fact that in urban areas development attracts further development. The model offers the possibility of predicting the global properties (such as scaling behavior) of urban morphologies.…”

“…This last modification corresponds to the percolation problem in the presence of a concentration gradient proposed in [21][22][23]. Here, the urban areas are red, and the external perimeter or urban boundary of the largest cluster connected to the CBD is light green.…”

“…Since p(r) decreases as one moves away from the core, the probability that the largest cluster remains connected decreases with r. The mean distance of the perimeter from the center r f is determined by the value of r for which p(r) equals the percolation threshold-i.e. p(r f ) = p c , so [21][22][23] …”

“…The width of the external perimeter is a function of the concentration gradient λ and it is known to scale as [21] …”

“…The fact that we are unable to see this limit might be due to numerical limitations near the mean field point α = 0. The number of sites of the frontier N f also scales with the concentration gradient [21] …”