April 19, 1996
The fractal properties of models of randomly placed $n$-dimensional spheres ($n$=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using analytical and numerical calculations it is shown that in the regime of low volume fraction occupied by the spheres, apparent fractal behavior is observed for a range of scales between physically relevant cut-offs. The width of this range, typically spanning between one and two orders of magnitude, is in very good agreement with the typical range observed in experimental measurements of fractals. The dimensions are not universal and depend on density. These observations are applicable to spatial, temporal and spectral random structures. Polydispersivity in sphere radii and impenetrability of the spheres (resulting in short range correlations) are also introduced and are found to have little effect on the scaling properties. We thus propose that apparent fractal behavior observed experimentally over a limited range may often have its origin in underlying randomness.
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January 28, 1997
We show that when the standard techniques for calculating fractal dimensions in empirical data (such as the box counting) are applied on uniformly random structures, apparent fractal behavior is observed in a range between physically relevant cutoffs. This range, spanning between one and two decades for densities of 0.1 and lower, is in good agreement with the typical range observed in experiments. The dimensions are not universal and depend on density. Our observations are a...
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January 7, 1998
The notion of the abundance of fractals is critically re-examined in light of surprising data regarding the scaling range in empirical reports on fractality.
February 1, 1994
In analogy to recent results on non-universal roughening in surface growth [Lam and Sander, Phys. Rev. Lett. {\bf 69}, 3338 (1992)], we propose a variant of diffusion-limited aggregation ($DLA$) in which the radii of the particles are chosen from a power law distribution. For very broad distributions, the huge particles dominate and the fractal dimension is calculated exactly using a scaling theory. For narrower distributions, it crosses back to DLA. We simulated $1200$ clust...
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A random sequential box-covering algorithm recently introduced to measure the fractal dimension in scale-free networks is investigated. The algorithm contains Monte Carlo sequential steps of choosing the position of the center of each box, and thereby, vertices in preassigned boxes can divide subsequent boxes into more than one pieces, but divided boxes are counted once. We find that such box-split allowance in the algorithm is a crucial ingredient necessary to obtain the fra...
March 12, 2002
We consider growing spheres seeded by random injection in time and space. Growth stops when two spheres meet leading eventually to a jammed state. We study the statistics of growth limited by packing theoretically in d dimensions and via simulation in d=2, 3, and 4. We show how a broad class of such models exhibit distributions of sphere radii with a universal exponent. We construct a scaling theory that relates the fractal structure of these models to the decay of their pore...
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The fractal dimension of a surface allows its degree of roughness to be characterised quantitatively. However, limited effort has been attempted to compute the fractal dimension of surfaces computed from precisely known atomic coordinates from computational biomolecular and nanomaterial studies. This work proposes methods to estimate the fractal dimension of the surface of any three-dimensional object composed of spheres, by representing it as either a voxelised point cloud o...
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Static structure factors are computed for large-scale, mechanically stable, jammed packings of frictionless spheres (three dimensions) and disks (two dimensions) with broad, power-law size dispersity characterized by the exponent $-\beta$. The static structure factor exhibits diverging power-law behavior for small wavenumbers, allowing us to identify a structural fractal dimension, $d_f$. In three dimensions, $d_f \approx 2.0$ for $2.5 \le \beta \le 3.8 $, such that each of t...
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