Branch order and ramification analysis of large diffusion-limited-aggregation clusters

Ossadnik, Peter

doi:10.1103/physreva.45.1058

Cited by 71 publications

(55 citation statements)

References 12 publications

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“…This is, however, a rather surprising conclusion considering that even a cursory visual inspection of the clusters would seem to suggest otherwise. There is also other convincing direct numerical evidence in the literature that supports the case for the self-similar nature of this branch structure [31][32][33]. This thus suggests that it is the implementation of the sandbox method that is at fault.…”

Section: B Evaluation Of D(q)mentioning

confidence: 61%

“…Since the inception of the model, DLA has proved especially difficult to characterize accurately because many of the measurements performed on clusters exhibit strong finite-size effects [18,21,30,32,[36][37][38][39][40]. Indeed, we have seen their presence in our own measurements of lacunarity and the spectrum of generalized dimensions D(q).…”

Section: Discussionmentioning

confidence: 86%

“…For if it was, then μ max and μ min would scale differently with , thus implying that there exist disparate regions in the clusters with quite different branch structure. Using the language of branch order analysis, this would require, for example, that either or both of the bifurcation and length ratios should differ with location within the cluster [31][32][33]. It is hard to see why this should be so.…”

Section: -8mentioning

confidence: 99%

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Multifractal analysis of the branch structure of diffusion-limited aggregates

Hanan

Heffernan

2012

Phys. Rev. E

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We examine the branch structure of radial diffusion-limited aggregation (DLA) clusters for evidence of multifractality. The lacunarity of DLA clusters is measured and the generalized dimensions D(q) of their mass distribution is estimated using the sandbox method. We find that the global n-fold symmetry of the aggregates can induce anomalous scaling behavior into these measurements. However, negating the effects of this symmetry, standard scaling is recovered.

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Section: B Evaluation Of D(q)mentioning

confidence: 61%

Section: Discussionmentioning

confidence: 86%

Section: -8mentioning

confidence: 99%

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Multifractal analysis of the branch structure of diffusion-limited aggregates

Hanan

Heffernan

2012

Phys. Rev. E

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“…Ossadnik (9) has considered the branching statistics of 47 off-lattice DLA clusters, each with 10 6 particles; a typical example is illustrated in Fig. 15.…”

Section: Dlamentioning

confidence: 99%

Symmetries in geology and geophysics

Turcotte

Newman

1996

Proc. Natl. Acad. Sci. U.S.A.

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Symmetries have played an important role in a variety of problems in geology and geophysics. A large fraction of studies in mineralogy are devoted to the symmetry properties of crystals. In this paper, however, the emphasis will be on scale-invariant (fractal) symmetries. The earth's topography is an example of both statistically self-similar and self-affine fractals. Landforms are also associated with drainage networks, which are statistical fractal trees. A universal feature of drainage networks and other growth networks is side branching. Deterministic space-filling networks with side-branching symmetries are illustrated. It is shown that naturally occurring drainage networks have symmetries similar to diffusion-limited aggregation clusters.Symmetries appear in a wide variety of contexts in geology and geophysics. Rocks are made up of minerals and the symmetry aspects of minerals constitute a major part of any course in mineralogy. Many advances in x-ray diffraction were motivated by studies of minerals. The symmetries of crystalline structures are generally associated with translations, rotations, and reflections. In this paper, however, we will focus our attention on a fourth symmetry: scale invariance. Scale invariance is generally associated with fractals and self-similarity. The classic example is the Cantor set, illustrated in Fig. 1. At each order, the remaining line segments are divided into three parts and two are retained. The fractal dimension is D ϭ log(N 2 ͞N 1 )͞ log(r 1 ͞r 2 ), where N is number and r is length for the Cantor set D ϭ log2͞log3 ϭ 0.6309, intermediate between the Euclidean dimension of a line (D ϭ 1) and a point (D ϭ 0).It should be remembered that Mandelbrot (1) introduced the concept of fractals in terms of the length of a rocky coastline. The length of a rocky coastline scales with the length of the measuring rod used as a fractional inverse power (2). This is a statistical symmetry rather than a deterministic symmetry. An illustration of this is given in Fig. 2. On the left is a third-order Koch triadic island. This is a deterministic fractal with D ϭ log4͞log3 ϭ 1.262, intermediate between the Euclidean dimension of an area (D ϭ 2) and a line (D ϭ 1). On the right is the map of an actual island, Dear Island, Maine. Using either the measuring-rod method or the box-counting method (2), this island satisfies fractal statistics to a good approximation, with D Ϸ 1.4. Rocky coastlines generally exhibit statistical scale-invariant symmetries. Other examples of scale-invariant symmetries in geology and geophysics include fragments, faults, earthquakes, ore deposits, oil fields, and volcanic eruptions. The fractal dimension associated with a scale-invariant symmetry is a quantitative measure of texture; increased fractal dimensions correspond to increased roughness.Self-similar symmetries can also be associated with selfaffine fractals; some statistically self-similar time series are examples of self-affine fractals. A deterministic, self-affine fractal is illustrated in Fig....

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“…Other examples include the finite-size effects on the noise magnitude [12], the branch subtending angle [13], and the radial density [14]. These of the clusters instead of the 6ne details, the potentials for the clusters are similar.…”

Section: Introductionmentioning

confidence: 99%

Finite-size effects in diffusion-limited aggregation

Lam

1995

Phys. Rev. E

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This paper discusses a large variety of numerical results on difFusion-limited aggregation (DLA) to support the view that asymptotically large DLA is self-similar and the scaling of the geometry can be speci6ed by the fractal dimension alone. Deviations from simple scaling observed in many simulations are due to 6nite-size effects. I explain the relationship between the 6nite-size effects in various measurements and how they can arise due to a crossover of the noise magnitude in the growth process. Complex scaling hypotheses including anomalous scaling of the width of the growing region, multiscaling of the cluster radial density, infinite drift of the e-neighborhood filling ratio, nonmultifractal scaling of the growth probability measure, and geometrical multifractality, are shown to lead to physically unacceptable predictions. PACS number(s): 61.43.Hv, 05.40.+j

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Branch order and ramification analysis of large diffusion-limited-aggregation clusters

Cited by 71 publications

References 12 publications

Multifractal analysis of the branch structure of diffusion-limited aggregates

Multifractal analysis of the branch structure of diffusion-limited aggregates

Symmetries in geology and geophysics

Finite-size effects in diffusion-limited aggregation

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