The formation of fractal objects

Abstract

During recent years there has been an increasing interest in fractal or dendritic structures in nature [1]. A familiar example here at Les Houches, where we are surrounded by some of the most beautiful mountains in the world, is the snowflake. While browsing through the school library, I found out that Kepler was probably the first well-known scientist to be interested in this subject. He wrote an essay [2[ in which he attempted to explain the sixfold symmetry of snowflakes. Although this symmetry has been well understood for some time, it is remarkable that the origin of the dendritic appearance of snowflakes has been elucidated only rather recently [3, 4]. In this talk I would like to discuss a simple stochastic model for the formation of fractal aggregates which was introduced by Witten and Sander [5]. It has the great advantage that it can be readily implemented on your personal computer to produce remarkable fractal structures like the example shown in fig. 1. This aggregate is formed in the following manner: Locate a seed particle fixed at the center of your screen, and introduce a second particle moving along a random walk starting at the edge of the screen. If this particle comes within a distance a from the seed particle it gets trapped. When this happens introduce another random walker which can now get stuck in the neighborhood of either of the two fixed particles. Repeat this process n times, and you get a structure like the one shown in fig. 1. It is easy to understand the qualitative behavior of this model. The tree-like structure is due to the fact that a random walker coming from outside has a high probability of getting trapped in the outer branches of the growing aggregate. Careful measurements show that this aggregate has remarkable scaling and universality properties. For example, the densitydensity correlation function C(r) has a simple power law behavior [5, 6] as a function of the distance, C(r) ~ r -A, where the power A depends on the dimensionality d. This result is reminiscent of the behavior of critical phenomena in statistical mechanics, and suggests the possible application of renormalization group techniques [7] to evaluate the exponent A or fractal dimension D = d A. I would like to report here on a more modest and old-fashioned approach, which was carried out in collaboration with S. Alexander, R. Ball, T.A. Witten, L.M. Sander and R. Richter. Consider the continuum version of the discrete stochastic model of Witten and Sander [5]. This formulation leads to a pair of nonlinear partial differential equations for the density p of the growing aggregate, and the density u of the diffusing particle. A remarkable fact is that for the case of spherical symmetry these equations can be solved analytically in an asymptotic regime [8, 9], which elucidates the universality and scaling properties of the model. The generalized continuum equations for diffusion and attachment take the form [8--11]