Laplacian Fractal Growth in Media with Quenched Disorder

We analyze the combined effect of a Laplacian field and quenched disorder for the generation of fractal structures with a study, both numerical and theoretical, of the quenched dielectric breakdown model (QDBM). The growth dynamics is shown to evolve from the avalanches of invasion percolation (IP) to the smooth growth of Laplacian fractals, i. e. diffusion limited aggregation (DLA) and the dielectric breakdown model (DBM). The fractal dimension is strongly reduced with respect to both DBM and IP, due to the combined effect of memory and field screening. This implies a specific relation between the fractal dimension of the breakdown structures (dielectric or mechanical) and the microscopic properties of disordered materials.Comment: 11 pages Latex (revtex), 3 postscript figures included. Submitted to PR

Generalized Dielectric Breakdown Model

We propose a generalized version of the Dielectric Breakdown Model (DBM) for generic breakdown processes. It interpolates between the standard DBM and its analog with quenched disorder, as a temperature like parameter is varied. The physics of other well known fractal growth phenomena as Invasion Percolation and the Eden model are also recovered for some particular parameter values. The competition between different growing mechanisms leads to new non-trivial effects and allows us to better describe real growth phenomena. Detailed numerical and theoretical analysis are performed to study the interplay between the elementary mechanisms. In particular, we observe a continuously changing fractal dimension as temperature is varied, and report an evidence of a novel phase transition at zero temperature in absence of an external driving field; the temperature acts as a relevant parameter for the ``self-organized'' invasion percolation fixed point. This permits us to obtain new insight into the connections between self-organization and standard phase transitions.Comment: Submitted to PR

Theory of Boundary Effects in Invasion Percolation

We study the boundary effects in invasion percolation with and without trapping. We find that the presence of boundaries introduces a new set of surface critical exponents, as in the case of standard percolation. Numerical simulations show a fractal dimension, for the region of the percolating cluster near the boundary, remarkably different from the bulk one. We find a logarithmic cross-over from surface to bulk fractal properties, as one would expect from the finite-size theory of critical systems. The distribution of the quenched variables on the growing interface near the boundary self-organises into an asymptotic shape characterized by a discontinuity at a value $x_c=0.5$, which coincides with the bulk critical threshold. The exponent $\tau^{sur}$ of the boundary avalanche distribution for IP without trapping is $\tau^{sur}=1.56\pm0.05$; this value is very near to the bulk one. Then we conclude that only the geometrical properties (fractal dimension) of the model are affected by the presence of a boundary, while other statistical and dynamical properties are unchanged. Furthermore, we are able to present a theoretical computation of the relevant critical exponents near the boundary. This analysis combines two recently introduced theoretical tools, the Fixed Scale Transformation (FST) and the Run Time Statistics (RTS), which are particularly suited for the study of irreversible self-organised growth models with quenched disorder. Our theoretical results are in rather good agreement with numerical data.Comment: 11 pages, 13 figures, revte

A perturbative approach to the Bak-Sneppen Model

We study the Bak-Sneppen model in the probabilistic framework of the Run Time Statistics (RTS). This model has attracted a large interest for its simplicity being a prototype for the whole class of models showing Self-Organized Criticality. The dynamics is characterized by a self-organization of almost all the species fitnesses above a non-trivial threshold value, and by a lack of spatial and temporal characteristic scales. This results in {\em avalanches} of activity power law distributed. In this letter we use the RTS approach to compute the value of $x_c$, the value of the avalanche exponent $\tau$ and the asymptotic distribution of minimal fitnesses.Comment: 4 pages, 3 figures, to be published on Physical Review Letter

Theory of Self-organized Criticality for Problems with Extremal Dynamics

We introduce a general theoretical scheme for a class of phenomena characterized by an extremal dynamics and quenched disorder. The approach is based on a transformation of the quenched dynamics into a stochastic one with cognitive memory and on other concepts which permit a mathematical characterization of the self-organized nature of the avalanche type dynamics. In addition it is possible to compute the relevant critical exponents directly from the microscopic model. A specific application to Invasion Percolation is presented but the approach can be easily extended to various other problems.Comment: 11 pages Latex (revtex), 3 postscript figures included. Submitted to Europhys. Let

Dynamics of Fractures in Quenched Disordered Media

We introduce a model for fractures in quenched disordered media. This model has a deterministic extremal dynamics, driven by the energy function of a network of springs (Born Hamiltonian). The breakdown is the result of the cooperation between the external field and the quenched disorder. This model can be considered as describing the low temperature limit for crack propagation in solids. To describe the memory effects in this dynamics, and then to study the resistance properties of the system we realized some numerical simulations of the model. The model exhibits interesting geometric and dynamical properties, with a strong reduction of the fractal dimension of the clusters and of their backbone, with respect to the case in which thermal fluctuations dominate. This result can be explained by a recently introduced theoretical tool as a screening enhancement due to memory effects induced by the quenched disorder.Comment: 7 pages, 9 Postscript figures, uses revtex psfig.sty, to be published on Phys. Rev.

Exact solution of diffusion limited aggregation in a narrow cylindrical geometry

The diffusion limited aggregation model (DLA) and the more general dielectric breakdown model (DBM) are solved exactly in a two dimensional cylindrical geometry with periodic boundary conditions of width 2. Our approach follows the exact evolution of the growing interface, using the evolution matrix E, which is a temporal transfer matrix. The eigenvector of this matrix with an eigenvalue of one represents the system's steady state. This yields an estimate of the fractal dimension for DLA, which is in good agreement with simulations. The same technique is used to calculate the fractal dimension for various values of eta in the more general DBM model. Our exact results are very close to the approximate results found by the fixed scale transformation approach.Comment: 18 pages RevTex, 6 eps figure

Renormalization group approach to the critical behavior of the forest fire model

We introduce a Renormalization scheme for the one and two dimensional Forest-Fire models in order to characterize the nature of the critical state and its scale invariant dynamics. We show the existence of a relevant scaling field associated with a repulsive fixed point. This model is therefore critical in the usual sense because the control parameter has to be tuned to its critical value in order to get criticality. It turns out that this is not just the condition for a time scale separation. The critical exponents are computed analytically and we obtain $\nu=1.0$, $\tau=1.0$ and $\nu=0.65$, $\tau=1.16$ respectively for the one and two dimensional case, in very good agreement with numerical simulations.Comment: 4 pages, 3 uuencoded Postcript figure

Growing Cayley trees described by Fermi distribution

We introduce a model for growing Cayley trees with thermal noise. The evolution of these hierarchical networks reduces to the Eden model and the Invasion Percolation model in the limit $T\to 0$, $T\to \infty$ respectively. We show that the distribution of the bond strengths (energies) is described by the Fermi statistics. We discuss the relation of the present results with the scale-free networks described by Bose statistics