Transition in the Fractal Properties from Diffusion Limited Aggregation to Laplacian Growth via their Generalization

Abstract

We study the fractal and multifractal properties (i.e. the generalized dimensions of the harmonic measure) of a 2-parameter family of growth patterns that result from a growth model that interpolates between Diffusion Limited Aggregation (DLA) and Laplacian Growth Patterns in 2-dimensions. The two parameters are βwhich determines the size of particles accreted to the interface, and C which measures the degree of coverage of the interface by each layer accreted to the growth pattern at every growth step. DLA and Laplacian Growth are obtained at β=0, C=0 and β=2, C=1, respectively. The main purpose of this paper is to show that there exists a line in the β-C phase diagram that separates fractal (D<2) from non-fractal (D=2) growth patterns. Moreover, Laplacian Growth is argued to lie in the non-fractal part of the phase diagram. Some of our arguments are not rigorous, but together with the numerics they indicate this result rather strongly. We first consider the family of models obtained for β=0, C>0, and derive for them a scaling relation D=2 * D_3. We then propose that this family has growth patterns for which D=2 for some C>C_{cr}, where C_{cr} may be zero. Next we consider the whole β-C phase diagram and define a line that separates 2-dimensional growth patterns from fractal patterns with D<2. We explain that Laplacian Growth lies in the region belonging to 2-dimensional growth patterns, motivating the main conjecture of this paper, i.e. that Laplacian Growth patterns are 2-dimensional. The meaning of this result is that the branches of Laplacian Growth patterns have finite (and growing) area on scales much larger than any ultra-violet cut-off length.