Showing papers by "Luciano Pietronero published in 1989"
TL;DR: In this paper, an intrinsic test for the validity of the concept of wedges in the modelling of the minimal singularity strength of the growth probability distribution in the dielectric breakdown model is presented.
Abstract: We present an intrinsic test for the validity of the concept of wedges in the modelling of the minimal singularity strength of the growth probability distribution in the dielectric breakdown model. The conclusion is that the wedge shape may be a reasonable approximation in some particular cases but it is not a concept of general validity.
2 citations
TL;DR: In this article, the main ideas of a new theoretical approach for irreversible fractal growth are discussed, which is substantially different from the renormalization group theory of equilibrium phase transitions.
1 citations
TL;DR: In this paper, a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on Laplace equation and provides a systematic method for the calculation of the fractal dimension is presented.
Abstract: We introduce a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on Laplace equation and provides a systematic method for the calculation of the fractal dimension. A specific application to the Dielectric Breakdown Model (including therefore DLA) in two dimension is presented. The analogies and differences with respect to the Renormalization Group are discussed.
TL;DR: In this paper, the authors introduced a new theoretical framework for 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.
01 Jan 1989
TL;DR: 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 and makes clear why the usual renormalization schemes are not very suitable for these problems.
Abstract: In order to understand the physical origin of 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 very suitable for these problems. Here we describe the basic ideas of this new approach and report about recent developments including the application to the fractal dimension of the percolating cluster interpreted as a problem of fractal growth.