Characterizing Fractal and Hierarchical Morphologies Beyond the Fractal Dimension

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.

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

1994 ◽  
Vol 367 ◽  
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.

QUANTIFYING MORPHOLOGY OF SCALE-INVARIANT STRUCTURES BEYOND THE FRACTAL DIMENSION

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 985-991
Author(s):  
RAPHAEL BLUMENFELD ◽  
ROBIN C. BALL
Keyword(s):  
Correlation Function ◽  
Fractal Dimension ◽  
Pair Correlation Function ◽  
Pair Correlation ◽  
Scale Invariant ◽  
Diffusion Limited Aggregation ◽  
Suitable Candidate ◽  
Diffusion Limited ◽  
Invariant Structures ◽  
Admissible Solutions

We present a correlation scheme to quantify the morphology beyond the standard fractal dimension which corresponds to information from the pair correlation function. The method consists of analyzing hierarchical correlations in log-space thus summing contributions from higher order correlations in the usual space coordinates. The scheme gives information on the characteristics of structure which can be used as a fingerprint to distinguish between structures with the same fractal dimension. The method is also sensitive to oscillations in logarithmic scales, which are admissible solutions for renormalisation equations. Such oscillations appear as resonances, thus making this scheme suitable to analyze such phenomena experimentally. Illustrative examples are given for all those applications by analyzing numerically grown structures. The case of the unrestricted Brownian walk is exactly calculated. We discuss an application of this scheme to check recent analytic results obtained for scale-invariant branching mechanisms in slow cracking patterns and in noise-reduced diffusion-limited-aggregation. We propose that this method is a suitable candidate to quantify the presently qualitative concept of morphology.

Asymptotic behavior and noise reduction in diffusion-limited aggregation models

1989 ◽  
Vol 39 (5) ◽  
pp. 2587-2592 ◽  
Author(s):  
T. Aukrust ◽  
M. A. Novotny ◽  
D. A. Browne ◽  
K. Kaski
Keyword(s):  
Asymptotic Behavior ◽  
Noise Reduction ◽  
Diffusion Limited Aggregation ◽  
Diffusion Limited

Observation of fractal patterns in C60-polymer thin films

1994 ◽  
Vol 9 (9) ◽  
pp. 2216-2218 ◽  
Author(s):  
H.J. Gao ◽  
Z.Q. Xue ◽  
Q.D. Wu ◽  
S. Pang
Keyword(s):  
Thin Films ◽  
Fractal Dimension ◽  
Long Range ◽  
Diffusion Limited Aggregation ◽  
Aggregation Model ◽  
Diffusion Limited ◽  
Range Correlation ◽  
Microscopic Features ◽  
Fractal Patterns ◽  
Cluster Diffusion

We report the observation of fractal patterns in C60-tetracyanoquinodimethane thin films. The fractal patterns and their microscopic features are described and characterized. The fractal dimension was determined to be 1.69 ± 0.07. According to the characterization results, the observed fractals are compared to the cluster-diffusion-limited-aggregation model. The growth of the fractal patterns in the thin films is also in terms of the existing long-range correlation.

Fractal dimension of the accessible perimeter of diffusion-limited aggregation

1989 ◽  
Vol 40 (3) ◽  
pp. 1713-1716 ◽  
Author(s):  
Cettina Amitrano ◽  
Paul Meakin ◽  
H. Eugene Stanley
Keyword(s):  
Fractal Dimension ◽  
Diffusion Limited Aggregation ◽  
Diffusion Limited

SCALING BEHAVIOR OF DIFFUSION LIMITED AGGREGATION IN PERCOLATION CLUSTER

2008 ◽  
Vol 22 (07) ◽  
pp. 507-513 ◽  
Author(s):  
QIANG TANG
Keyword(s):  
Fractal Dimension ◽  
Computer Model ◽  
Percolation Cluster ◽  
Multifractal Spectrum ◽  
Scaling Behavior ◽  
Diffusion Limited Aggregation ◽  
Occupancy Probability ◽  
Diffusion Limited ◽  
Multifractal Spectra ◽  
Simulation Results

This paper presents a computer model of diffusion limited aggregation (DLA) in percolation cluster. Simulation of the aggregation clusters in percolation cluster with varying occupancy probability is performed, and their fractal dimension and multifractal spectrum are obtained. The simulation results show that the percolation cluster has stronger effects on the aggregation clusters' pattern structure when occupancy probability is smaller. The dimension Df of aggregation clusters increases together with the increase of occupancy probability. Furthermore, the multifractal spectra f(α) curve becomes higher and the range of singularity α wilder. The bigger the occupancy probability is, the more irregular and non-uniform the aggregation clusters becomes.

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