Fractal and Multifractal Structures in Kinetic Critical Phenomena

Abstract

Kinetic Critical Phenomena consist of irreversible growth models that generate fractal structures. In the last few years a large amount of activity has been devoted to these problems that represent one of the most challenging fields in today’s theoretical physics. The idea is to understand the physical origin for the development of the many fractal structures one can observe in nature. In this respect one of the most interesting models is the one formulated to describe the patterns of dielectric breakdown. It is based on a combination of Laplace equation and a probability field that give rise to random fractals. An essential element for the understanding of these results is the proper characterization of the self-similar properties of the growth probability field. To this purpose the more general concept of multifractal is necessary. This is discussed in some detail and applied to the above model showing that the growth probability should be described by a continuous distribution of singularities. The origin of multifractality is then elucidated by showing that, despite its apparent complexity, it arises in a natural way even in simple random multiplicative processes. This allows to consider apparently unrelated problems like the properties of self-similarity of Anderson localized wavefunctions for which a complete characterization is given. The main open problems as well as the perspectives for future development are briefly discussed for each topic.