Fixed Scale Transformation Approach Research Articles

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.

Theory of extremal dynamics with quenched disorder: Invasion percolation and related models.

The study of phenomena such as capillary displacement in porous media, fracture propagation, and interface dynamics in quenched random media has attracted a great deal of interest in the last few years. This class of problems does not seem to be treatable with the standard theoretical methods, and the only analytical results come from scaling theory or mapping, for some of their properties, to other solvable models. In this paper a recently proposed approach to problems with extremal dynamics in quenched disordered media, named run time statistics (RTS) or quenched-stochastic transformation, is described in detail. This method allows us to map a quenched dynamics such as invasion percolation onto a stochastic annealed process with cognitive memory. By combining RTS with the fixed scale transformation approach, we develop a general and systematic theoretical method to compute analytically the critical exponents of invasion percolation, with and without trapping, and directed invasion percolation. In addition we can also understand and describe quantitatively the self-organized nature of the process. \textcopyright{} 1996 The American Physical Society.

The fixed scale transformation approach to spatiotemporal intermittency in coupled map lattices

A one dimensional array of nonlinear maps with Laplacian couplings is studied at the phase transition between the asymptotically “turbulent” and “laminar” phases. Along the critical boundary, where spatiotemporal intermittency is observed, turbulent sites form a fractal set. The fixed scale transformation approach allows us to simultaneously determine the fractal dimension of the turbulent sites and the invariant asymptotic measure. Defining scale invariant conditional probabilities, and taking into account the nearest neighbour correlations exactly, we determine an asymptotic distribution function for the variables pertaining to sites on or neighbouring the turbulent clusters at any given level of coarse graining and obtain the fractal dimension analytically.

Fixed-scale transformation approach to linear and branched polymers

The radius exponent of two- and three-dimensional self-avoiding walks and branched polymers are computed in the fixed-scale transformation framework. The method requires the knowledge of the critical fugacity kc, but from this non-universal parameter it is possible to compute the universal critical exponent. The results obtained are within 1% of exact or numerical values. This confirms the versatility and quantitative power of this new theoretical approach and gives the opportunity to provide a discussion of the analogies and differences between the real space renormalization group and the fixed-scale transformation method.

Asymptotic screening in the scale invariant growth rules for Laplacian fractals

A key element in the fixed scale transformation approach to fractal growth is the use of the asymptotic scale invariant dynamics of the growth process. This is a non-universal element, analogous to the critical probability or temperature in percolation or Ising problems. The essential property to generate fractal structure is the persistence of screening effects in the asymptotic regime. To investigate this problem we use a renormalization procedure in which the noise reduction parameter is the critical one. The approach is based on the growth process itself and shows a non-trivial fixed point where the screening properties are preserved. This result guarantees the existence of the asymptotic fractal structure and clearly defines the basic elements of the growth rules used in the fixed scale transformation method.

Fixed scale transformation approach to critical fluctuations and fractal growth

Abstract A geometrical approach to the study of critical phenomena calls for a fractal description of critical fluctuations. The fixed scale transformation (FST) approach introduces such a description, incorporating both the scale and translational invariance properties of the critical state. Besides providing a theoretical understanding of the genesis of fractal structures, it enables one to compute the fractal dimension of the critical clusters to a high degree of accuracy, in terms of the critical statistics of the underlying model. Laplacian models of irreversible fractal growth, to which the FST approach was first applied, are presented within the general framework of critical phenomena. This geometrical approach links the growth rules, which are fully nonlocal in both space and time (measured in number of particles added), to the morphology of the grown cluster and its scaling behaviour. Possible directions towards a more comprehensive theory of fractal growth are discussed.

Fixed scale transformation applied to cluster-cluster aggregation in two and three dimensions

Abstract Recently it has been introduced a new theoretical framework named fixed scale transformation (FST), which appears particularly suitable to study the growth of fractal structures. This method allows the first analytical study of the process of cluster-cluster aggregation (CCA). The FST approach can in fact be generalized in a natural and relatively simple way to the case of CCA. Here we present detailed results for teh analytical calculation of the fractal dimension of the aggregates. For CCA in two dimensions the computed value is D = 1.39 and in three dimensions is D = 1.9, to be compared with the simulation results that are respectively D = 1.45 and D = 1.9. Furthermore the approximation scheme of the FST can be implemented in a systematic way to estimate quantitatively higher order corrections and to study variation of the original model. This application is of particular relevance because CCA has eluded all the standard theoretical approach and in particular it cannot even be formulated from the point of view of renormalization group methods.

Fixed scale transformation approach to the multifractcal properties of the growth probabilities in the dielectric breakdown model

The separation of the properties of the growth probability distribution in two different contributions, as discussed in the previous paper, corresponds naturally to the approximation scheme of the fixed scale transformation (FST) method. The growth probabilities used to compute the FST matrix elements represent the essential elements of the multiplicative process that gives rise to the regular part (the only one relevant to the growth process) of the multifractal spectrum. The FST uses these probabilities directly without the need of introducing a multifractal spectrum explicitly. This, however, can be obtained as a by-product of the FST method. We present here analytical calculations for the regular part of the multifractal spectrum of the dielectric breakdown model with different values of the parameter ν. The results are good for η ⩾ 1 and less accurate for η < 1. In fact for small η values, and in order to recover the Eden limit, it is necessary to go to higher order and possibly to include self-affine properties explicitly.

Fixed scale transformation approach to the nature of relaxation clusters in self-organized criticality.

We use the fixed scale transformation method, developed for fractal growth, to investigate analytically the nature of clusters in self-organized criticality (SOC). In two dimensions the clusters of sites involved in a relaxation process turn out to be compact (D=2) because of the absence of effective screening. Therefore they are more similar to Eden-type clusters (possibly with a rough surface) than to those of the usual fractal growth models. This result is in good agreement with the computer simulations and one can conjecture that it should hold for any dimensions. The critical state corresponding to SOC dynamics is therefore of much simpler nature with respect to those of the usual fractal growth models.

Fixed scale transformation for Ising and Potts clusters

The fractal dimension of Ising and Potts clusters are determined via the fixed scale transformation approach, which exploits both the self-similarity and the dynamical invariance of these systems at criticality. The results are easily extended to droplets. A discussion of the interrelationships between the present approach and renormalization group methods as well as Glauber-type dynamics is provided.

Fractal dimension of intersection sets in the dielectric breakdown model

Clusters grown with the dielectric breakdown model (DBM) in the cylinder geometry show two growth phases: a scaling regime for cluster heights smaller than the cylinder circumference and a subsequent steady state, which is translational invariant in the main growth direction. The box-counting dimension of one-dimensional intersection sets of the clusters is studied for six different values of the growth parameter eta and four cylinder circumferences. The author finds that in the scaling regime this dimension depends on the height at which the intersection is made and the clusters are thus inhomogeneous. The results also show that clusters in the steady state are translational invariants in the main growth direction and self-similar in the direction perpendicular to it. A comparison between his findings and the theoretical results obtained from the fixed scale transformation approach by Pietronero et al. (1988), shows good agreement.