Finite-size effects in diffusion-limited aggregation

Abstract

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

Fractal Analysis of Aggregates of Non-Uniformly Sized Particles: An Application to Macaque Monkey Cortical Pyramidal Neurons

“…The cumulative mass method is particularly popular in applications because it can provide reliable estimates of fractal dimensions for smaller cluster sizes than the box-counting dimension. The cumulative mass method has been used extensively in the fractal analysis of clusters grown using computer growth models such as diffusion-limited aggregation 1,2,3,4 , in which the growing aggregates are composed of identical sized particles. The cumulative mass method has also been employed in the fractal analysis of neuronal morphology 5,6,7,8 .…”

Multifractal analysis of the branch structure of diffusion-limited aggregates

“…In both cases, we believe that their existence is a product of the randomness (or noise) inherent in DLA. More specifically, we shall argue that they are a consequence of the decrease in the relative noise magnitude with cluster growth [39,41]. This is the very same argument put forward by Lam to explain many of the complex scaling hypotheses purported for DLA [41].…”

“…More specifically, we shall argue that they are a consequence of the decrease in the relative noise magnitude with cluster growth [39,41]. This is the very same argument put forward by Lam to explain many of the complex scaling hypotheses purported for DLA [41]. In his paper, Lam discusses this idea in some depth, so we shall only give it the briefest of descriptions here.…”

“…We believe that the finite-size effects in our results reflect the slow decrease in the relative noise magnitude with growth. This argument has previously been put forward as an explanation for finite-size effects exhibited by other measured quantities in DLA [39,41].…”

“…However, there exist both numerical results and sound theoretical arguments to suggest that, as clusters grow larger, the relative noise magnitude decays to some nonzero value [39,41]. That is, there will always remain some randomness and inhomogeneity in DLA clusters.…”