Theory of fractal growth

Abstract

The problem we consider is to understand why nature gives rise to fractal structures. The first step is to formulate models of fractal growth based on physical mechanisms like the Diffusion Limited Aggregation and the more general Dielectric Breakdown Model. They are based on a simple iterative process governed by the Laplace equation and a stochastic field and they give rise to patterns that spontaneously evolve into random fractal structures of great complexity. In addition one would like to achieve a theoretical understanding of these models similar to that provided by the Renormalization Group for Ising-type models. Recently we have introduced a new theoretical framework for intrinsically critical growth models. This method is based on a Fixed Scale Transformation (with respect to the dynamical evolution) that defines a functional iteration for the distribution of elementary configurations that appear in a coarse graining process. This allows to include screening effects in terms of convergent series and to describe the intrinsic fluctuations of the boundary conditions. This approach clarifies the origin of fractal structures in these models and provides a systematic method for the calculation of the fractal dimension and the multifractal properties. It also makes clear why the usual renormalization schemes are not suitable for these problems. Here we describe the basic ideas of this new approach and report about recent developments, including the application to the percolation cluster interpreted as a problem of fractal growth.