Showing papers by "Luciano Pietronero published in 1988"

Theory of fractal growth

Abstract: In the past few years a great deal of activity has been devoted to the study of fractal structures [3] in relation to physical phenomena [4,5]. The prototype fractal growth model is based on a combination of the Laplace equation and a stochastic field. The first model of this class to be formulated was Diffusion Limited Aggregation (DLA) [6]. A few years later the more general Dielectric Breakdown Model (DBM) [7] was introduced. This model used the relation between the random walk and potential theory and made clear that growth could also occur “from inside”. In addition to their intrinsic theoretical interest, these models are now believed to capture the essential features necessary to describe pattern formation in seemingly different phenomena like electrochemical deposition, deudritic growth, dielectric breakdown, viscous fingering in fluids, fracture propagation and others [4,5].

Theory of Laplacian fractals: diffusion limited aggregation and dielectric breakdown model

Abstract: We describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the Laplace equation and a stochastic field. This new theory provides a systematic method for the calculation of the fractal dimension D and of the multifractal spectrum of the growth probability (ƒ(α)). A detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented. Our approach exploits the scale invariance of the Laplace equation that implies that the structure is self-similar both under growth and scale transformation. This allows one to introduce a Fixed Scale Transformation (instead of coarse graining as in the renormalization group theory) that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process. For the calculation of the matrix elements of this transformation one has to consider an infinite, but rapidly convergent, number of processes that occurs outside a considered diagram.

From physical dielectric-breakdown to the stochastic fractal model

Abstract: We show how the stochastic growth model we have formulated for dielectric breakdown is related to microscopic mechanisms. Beginning with gas discharges we focus on the origin of the stochastic features and the dependence of growth probability on the local electric field. For dielectric breakdown in solid insulators we argue that quenched disorder does not appreciably modify the patterns which appear in the gaseous case.

Exponentiated random walks, supersymmetry and localization

Abstract: We examine the nature and properties of the “exponentiated random walk” one-dimensional wavefunction Ψ0=exp[−x(x)], previously introduced in the context of the supersymmetric mappings of a classical Langevin random field problem. Three main results are presented. The first is that the state Ψ0 is extended, although it is the exact groundstate of a disordered one-dimensional quantum problem. The second is that in that problem supersymmetry is neither truly unbroken, or truly broken, we call this a situation of marginal unbroken supersymmetry and identify a class of other problems with the same property. The third result is obtained by studying the local behaviour of the wave function Ψ0 by means of generalized Lyapunov exponents. Locally, Ψ0 exhibits exponential localization, with a localization length identical to that of weak localization in the 1-dimensional Anderson problem.