Asymptotic behavior and noise reduction in diffusion-limited aggregation models

The equivalence between diffusion-limited aggregation (DLA) and growth in a Laplacian field is exploited to construct an algorithm for the simulation of DLA at large finite noise reduction. This algorithm allows performing of the limit towards infinite noise reduction, yielding a feasible prescription for the simulation of the noiseless, i.e., deterministic limit. Contrary to previous expectations as well as explicit predictions from an analytic theory, clusters grown without noise develop sidebranches. An explanation of this result in terms of the outer radius used in DLA simulations is suggested. Various implications, including the question whether or when noise reduction may accelerate the approach to asymptotic behavior, are discussed.

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Characterizing Fractal and Hierarchical Morphologies Beyond the Fractal Dimension

MRS Proceedings ◽

10.1557/proc-367-367 ◽

1994 ◽

Vol 367 ◽

Author(s):

Raphael Blumenfeld ◽

Robin C. Ball

Keyword(s):

Asymptotic Behavior ◽

Fractal Dimension ◽

Finite Size ◽

Size Analysis ◽

Corrections To Scaling ◽

Diffusion Limited Aggregation ◽

Diffusion Limited ◽

Markovian Processes ◽

Branching Patterns ◽

Admissible Solutions

AbstractWe present a novel correlation scheme to characterize the morphology of fractal and hierarchical patterns beyond traditional scaling. The method consists of analysing correlations between more than two-points in logarithmic coordinates. This technique has several advantages: i) It can be used to quantify the currently vague concept of morphology; ii) It allows to distinguish between different signatures of structures with similar fractal dimension but different morphologies already for relatively small systems; iii) The method is sensitive to oscillations in logarithmic coordinates, which are both admissible solutions for renormalization equations and which appear in many branching patterns (e.g., noise-reduced diffusion-limited-aggregation and bronchial structures); iv) The methods yields information on corrections to scaling from the asymptotic behavior, which is very useful in finite size analysis. Markovian processes are calculated exactly and several structures are analyzed by this method to demonstrate its advantages.

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

MRS Proceedings ◽

10.1557/proc-367-73 ◽

1994 ◽

Vol 367 ◽

Cited By ~ 1

Author(s):

B.B. Mandelbrot ◽

A. Vespignani ◽

H. Kaufman

Keyword(s):

Asymptotic Behavior ◽

Asymptotic Properties ◽

Finite Size ◽

Scaling Behavior ◽

Self Similarity ◽

Diffusion Limited Aggregation ◽

One Dimensional ◽

Diffusion Limited ◽

Large Numbers ◽

The One

AbstractIn 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.

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