The branching structure of diffusion-limited aggregates

– I analyze the topological structures generated by diffusion-limited aggregation (DLA), using the recently developed “branched growth model”. The computed bifurcation number B for DLA in two dimensions is B ≈ 4.9, in good agreement with the numerically obtained result of B ≈ 5.2. In high dimensions, B → 3.12; the bifurcation ratio is thus a decreasing function of dimensionality. This analysis also determines the scaling properties of the ramification matrix, which describes the hierarchy of branches. Nature creates an astonishing variety of branched structures. Some of these structures, such as trees, are created by biological systems; others, such as river networks, are created principally by physical phenomena. The simplest and best-understood physical model for the formation of branched structures is diffusion-limited aggregation (DLA), introduced over fifteen years ago [1]. The complex fractal structures generated by this model are seen in natural systems whose growth is controlled by diffusive processes [2]. Physicists have concentrated on understanding and predicting the fractal and multifractal properties of DLA [3, 4]. However, several authors have advanced an alternative approach, focussing on the topological self-similarity of DLA clusters, as measured by quantities typically used to describe river networks and other branched structures [5, 6]. This Letter is devoted to the computation of these quantities, using the “branched growth model” which this author and his collaborators have exploited to compute a number of properties of DLA [7]. Since this model is based on an analysis of the competition of branches in a hierarchical structure, it is peculiarly suited to the computation of topological quantities related to that structure. The growth rule for DLA can be defined inductively: introduce a random walker at a large distance from an n particle cluster, which sticks irreversibly at its point of first contact with the cluster, thereby forming the n + 1 particle cluster. Clusters grown in this way have an intricate branched structure, in which prominent branches screen internal regions of the cluster, preventing them from growing further. In addition, they contain no loops, since the particle closing a loop would have to attach to two pre-existing particles. The scaling of the radius of the cluster r with the number of particles n determines the fractal dimension D of Typeset using EURO-LTEX 2 EUROPHYSICS LETTERS Fig. 1. – A rooted tree, showing the Strahler indices and parent Strahler indices of each branch. The first number is the Strahler index of a branch, and the second is the Strahler index of the branch it meets when it loses its identity. The total number of index 1 branches is N1 = 10, of index 2 branches is N2 = 3, and of index 3 branches is N3 = 1. the cluster, n ∝ r. A number of quantities have been developed to describe the topological properties of river networks, which like DLA clusters have no loops. The most fundamental such quantity was introduced by Strahler, as a refinement of a scheme proposed by Horton [8]. Consider a rooted tree structure. The leaves of the structure are assigned the Strahler index i = 1. The index i of any other branch, which bifurcates into two branches with Strahler indices i1 and i2, is determined by