Theory of Fractal Growth

Localization of growth sites in diffusion-limited-aggregation clusters: Multifractality and multiscaling

“…One possibility to express ρ A (x, M) in terms of N(α, x, M) is by using Eqs. (5) and (8). Integration of Eq.…”

“…The growth of a diffusion limited aggregation (DLA) [1][2][3][4][5][6][7][8][9][10][11][12][13][14] cluster with mass M is described by the set of growth probabilities {p i } [15][16][17][18] where p i is the probability that perimeter site i will be the next to grow. One way to analyze the set {p i } is by calculating the "growth-probability distribution" n(α, M), where n(α, M)dα is the number of perimeter sites with α ≤ α i ≤ α + dα,…”

Growth and forms of Laplacian aggregates

“…It is known that the rich, fractal structure of DLA aggregates arises from the tip-splitting instability which is inferred from numerical and analytical calculations. Phenomenological schemes reproduce some of the most conspicuous global features, like the fractal dimension [2,3].…”

Disorderly Cluster Growth

“…Despite remarkable recent progress [7][8][9][10][11][12][13][14], no genuine understanding of DLA has emerged. There have been attempts to measure and understand the distribution of pi, and its lower cutoff Pmin, as well as to understand the the discovery of a "phase transition" in the behavior of the moments of the distribution, and the connection between multifractality and multiscaling [8][9][10][11][12][13][14].…”

“…To gain insight into the conditions under which multiscaling arises, consider the joint distribution function N (c~, x, M) (7) However, N (a, x, M) provides additional information about the location of growth sites with a specific a value which is not contained in n (a,M)…”