Theoretical concepts fpr fractal growth

Physical Review E, 2002

We study the fractal and multifractal properties (i.e. the generalized dimensions of the harmonic measure) of a 2-parameter family of growth patterns that result from a growth model that interpolates between Diffusion Limited Aggregation (DLA) and Laplacian Growth Patterns in 2-dimensions. The two parameters are β which determines the size of particles accreted to the interface, and C which measures the degree of coverage of the interface by each layer accreted to the growth pattern at every growth step. DLA and Laplacian Growth are obtained at β = 0, C = 0 and β = 2, C = 1, respectively. The main purpose of this paper is to show that there exists a line in the β − C phase diagram that separates fractal (D < 2) from non-fractal (D=2) growth patterns. Moreover, Laplacian Growth is argued to lie in the non-fractal part of the phase diagram. Some of our arguments are not rigorous, but together with the numerics they indicate this result rather strongly. We first consider the family of models obtained for β = 0, C > 0, and derive for them a scaling relation D = 2D3. We then propose that this family has growth patterns for which D = 2 for some C > Ccr, where Ccr may be zero. Next we consider the whole β − C phase diagram and define a line that separates 2-dimensional growth patterns from fractal patterns with D < 2. We explain that Laplacian Growth lies in the region belonging to 2-dimensional growth patterns, motivating the main conjecture of this paper, i.e. that Laplacian Growth patterns are 2-dimensional. The meaning of this result is that the branches of Laplacian Growth patterns have finite (and growing) area on scales much larger than any ultra-violet cut-off length.

2007

We study, both with numerical simulations and theoretical methods, a cellular automata model for continuum equations describing growth processes in the presence of an external flux of particles. As a result of local instabilities we find a fractal regime of growth for small external fluxes. The growing tip is selected with probability proportional to the curvature in the point. A parameter p gives the probability of lateral growth on the tip. The value of p determines the fractal dimension of the aggregate. Furthermore, for each value of p a cross-over between two different fractal dimensions is observed. Instead, the roughness exponent χ of the aggregates does not depend on p (χ ≃ 0.5). Fixed scale transformation approach is applied to compute theoretically the fractal dimension for one of the branches of the structure.

Physica A: Statistical Mechanics and its Applications, 1990

We consider percolation in two dimensions as a fractal growth problem, and apply to it the theory of fractal growth based on the fixed-scale transformation approach developed for diffusion-limited aggregation and the dielectric breakdown model. This represents an important test for this new theoretical method based on an additional invariance property with respect to the renormalization group. We compute the fractal dimension of the percolating cluster including terms up to third order. The result is D = 1.8830 for the square lattice and D = 1.8650 for the triangular lattice. These values are in excellent agreement with the universal exact result D = 91/48 = 1.8958 and also show the potential of this new method for standard problems.

Lecture Notes in Physics, 1994

We consider the fragmentation process with mass loss and discuss self-similar properties of the arising structure both in time and space focusing on dimensional analysis. This exhibits a spectrum of mass exponents θ, whose exact numerical values are given for which x −θ or t θz has the dimension of particle size distribution function c(x, t) where z is the kinetic exponent. We also give explicit scaling solution for special case. Finally, we identify a new class of fractals ranging from random to non-random and show that the fractal dimension increases with increasing order and a transition to strictly self-similar pattern occurs when randomness is completely seized. *

Physical Review E, 1994

In analogy to recent results on nonuniversal roughening in surface growth [Lam and Sander, Phys. Rev. Lett. 69, 3338 (1992)],we propose a variant of difFusion-limited aggregation (DLA) in which the radii of the particles are chosen from a power-law distribution. For very broad distributions, the huge particles dominate and the fractal dimension is calculated exactly using a scaling theory. For narrower distributions, it crosses back to DLA. We simulated 1200 clusters containing up to 200000 particles. The &actal dimensions obtained are in reasonable agreement with our theory. This variant of DLA might have relevance to the cluster-cluster aggregation model. PACS number(s): 64.60.Ak Diffusion-limited aggregation (DLA) [1] is an important prototype of fractal growth [2]. In two dimensions,

Physica Scripta, 1989

An overview is given of the known relations between various dynamical properties of fractal structure" such as vibration density, diffusion and electrical resistivity. Examples are given from the family of walk-generated fractals: SAW with and without bridges, /<-tolerant walks and randorn walks.

International Journal of Theoretical Physics, 2015

The fractals are obtained by using the model of diffusion-limited aggregation (DLA) for 40 ≤ L ≤ 240. The two-dimensional Ising model is simulated on the Creutz cellular automaton for 40 ≤ L ≤ 240. The critical exponents and the fractal dimensions are computed to be β = 0.124(8), γ = 1.747(10), α = 0.081(21), δ = 14.994(11), η = 0.178(10), ν = 0.960(23) and d Keywords Ising model • Finite-size scaling • Cellular automaton • Fractals

Physical Review E, 2004

The known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes. After that, we propose a description developed in the context of the intrinsic metric of fractals, which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime. We show that our approach has a stronger physical justification than previous works on this field. The most important result we present is the introduction of a dependence on time and space for the conductivity in fractals, which is deduced by scaling arguments and supported by computer simulations. Finally, the diffusion equation is used to introduce the possibility of reaction-diffusion processes on fractals and analyze their properties. Specifically, an analytic expression for the speed of the corresponding travelling fronts, which can be of great interest for application purposes, is derived.

International Journal of Computer Applications, 2012

The term fractal was coined in 1975 by Benoit Mandelbrot, from the Latin fractus, meaning "broken" or "fractured". In colloquial usage, a fractal is a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as "infinitely complex". Researchers used feedback systems to implement of a new iterative approach in the study of fractal models. The purpose of this paper is to present a review of literature in fractal analysis in recent years. In this review paper we have studied the work of various researchers in recent years on fractals models