Theoretical concepts fpr fractal growth

Abstract

After the introduction of fractal geometry by Benoit Mandelbrot the key problem is to understand why nature gives rise to fractal structures. This implies the formulation of models of fractal growth based on physical phenomena and the subsequent understanding of their mathematical structure in the same sense as the renormalization group has allowed to understand sing-type models. The models of diffusion-limited aggregation and the more general dielectric breakdown model, based on iterative processes governed by the Laplace equation and a stochastic field, have a clear physical meaning and they spontaneously evolve into random fractal structures of great complexity. From a theoretical point of view however it is not possible to describe them within usual concepts. Recently we have introduced a new theoretical framework for this class of problems. This 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. Here we summarize the basic ideas of this new approach and report about recent developments.