2 research outputs found
Iterated Conformal Dynamics and Laplacian Growth
The method of iterated conformal maps for the study of Diffusion Limited
Aggregates (DLA) is generalized to the study of Laplacian Growth Patterns and
related processes. We emphasize the fundamental difference between these
processes: DLA is grown serially with constant size particles, while Laplacian
patterns are grown by advancing each boundary point in parallel, proportionally
to the gradient of the Laplacian field. We introduce a 2-parameter family of
growth patterns that interpolates between DLA and a discrete version of
Laplacian growth. The ultraviolet putative finite-time singularities are
regularized here by a minimal tip size, equivalently for all the models in this
family. With this we stress that the difference between DLA and Laplacian
growth is NOT in the manner of ultraviolet regularization, but rather in their
deeply different growth rules. The fractal dimensions of the asymptotic
patterns depend continuously on the two parameters of the family, giving rise
to a "phase diagram" in which DLA and discretized Laplacian growth are at the
extreme ends. In particular we show that the fractal dimension of Laplacian
growth patterns is much higher than the fractal dimension of DLA, with the
possibility of dimension 2 for the former not excluded.Comment: 13 pages, 12 figures, submitted to Phys. Rev.