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The Geometry of Dla: Different Aspects of the Departure from Self-Similarity

Published online by Cambridge University Press:  03 September 2012

B.B. Mandelbrot ,
A. Vespignani  and
H. Kaufman
B.B. Mandelbrot
Affiliation:
Department of Mathematics, Yale University, New Haven, CT-06520-8283, USA
A. Vespignani
Affiliation:
Department of Mathematics, Yale University, New Haven, CT-06520-8283, USA
H. Kaufman
Affiliation:
Department of Mathematics, Yale University, New Haven, CT-06520-8283, USA
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Abstract

In order to understand better the morphology and the asymptotic behavior in Diffusion Limited Aggregation (DLA), we studied a large numbers of very large off-lattice circular clusters. We inspected both dynamical and geometric asymptotic properties, namely the moments of the particle's sticking distances and the scaling behavior of the transverse growth crosscuts, i.e., the one dimensional cuts by circles. The emerging picture for radial DLA departs from simple self-similarity for any finite size. It corresponds qualitatively to the scenario of infinite drift starting from the familiar five armed shape for small sizes and proceeding to an increasingly tight multi-armed shape. We show quantitatively how the lacunarity of circular clusters becomes increasingly “compact” with size. Finally, we find agreement among transverse cuts dimensions for clusters grown in different geometries, suggesting that the question of universality is best addressed on the crosscut.

Type
Research Article
Information
MRS Online Proceedings Library (OPL) , Volume 367: Symposium P – Fractal Aspects of Materials , 1994 , 73
Copyright
Copyright © Materials Research Society 1995

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