Anders Levermann

Laplacian Growth and Diffusion Limited Aggregation: Different Universality Classes

It had been conjectured that diffusion limited aggregates and Laplacian growth patterns (with small surface tension) are in the same universality class. Using iterated conformal maps we construct a one-parameter family of fractal growth patterns with a continuously varying fractal dimension. This family can be used to bound the dimension of Laplacian growth patterns from below. The bound value is higher than the dimension of diffusion limited aggregates, showing that the two problems belong to two different universality classes.

Multifractal structure of the harmonic measure of diffusion-limited aggregates

The method of iterated conformal maps allows one to study the harmonic measure of diffusion-limited aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as 10(-35), and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions D(q) are infinite for q<q*, where q* is of the order of -0.2. In the language of f(alpha) this means that alpha(max) is finite. The f(alpha) curve loses analyticity (the phenomenon of phase transition) at alpha(max) and a finite value of f(alpha(max)). We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of q* and f(alpha(max)). We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.

Quasistatic fractures in brittle media and iterated conformal maps

We study the geometrical characteristic of quasistatic fractures in brittle media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lamé equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern, it crosses over from about 0.5 for small scales to about 0.75 for large scales. We propose that this crossover reflects the increased ramification of the fracture pattern.

Convergent calculation of the asymptotic dimension of diffusion limited aggregates: Scaling and renormalization of small clusters

Diffusion limited aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (2-3 particles) to asymptotically large (n-->infinity) clusters. The computed dimension is D=1.713+/-0.003.

Quasistatic brittle fracture in inhomogeneous media and iterated conformal maps: Modes I, II, and III

The method of iterated conformal maps is developed for quasistatic fracture of brittle materials, for all modes of fracture. Previous theory, that was relevant for mode III only, is extended here to modes I and II. The latter require the solution of the bi-Laplace rather than the Laplace equation. For all cases we can consider quenched randomness in the brittle material itself, as well as randomness in the succession of fracture events. While mode III calls for the advance (in time) of one analytic function, modes I and II call for the advance of two analytic functions. This fundamental difference creates different stress distribution around the cracks. As a result the geometric characteristics of the cracks differ, putting mode III in a different class compared to modes I and II.

Bi-Laplacian Growth Patterns in Disordered Media

Experiments in quasi-two-dimensional geometry (Hele-Shaw cells) in which a fluid is injected into a viscoelastic medium (foam, clay, or associating polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a prerequisite to describing the fracture of elastic material is the solution of the bi-Laplace rather than the Laplace equation. In this Letter we close this gap, offering a theory of bi-Laplacian growth patterns based on the method of iterated conformal maps.

Transition in the fractal properties from diffusion-limited aggregation to Laplacian growth via their generalization

We study the fractal and multifractal properties (i.e., the generalized dimensions of the harmonic measure) of a two-parameter family of growth patterns that result from a growth model that interpolates between diffusion-limited aggregation (DLA) and Laplacian growth patterns in two dimensions. The two parameters are beta that determines the size of particles accreted to the interface, and C that 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 beta=0, C=0 and beta=2, C=1, respectively. The main purpose of this paper is to show that there exists a line in the beta-C phase diagram that separates fractal (D<2) from nonfractal (D=2) growth patterns. Moreover, Laplacian growth is argued to lie in the nonfractal 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 beta=0, C>0, and derive for them a scaling relation D=2D(3). We then propose that this family has growth patterns for which D=2 for some C>C(cr), where C(cr) may be zero. Next we consider the whole beta-C phase diagram and define a line that separates two-dimensional growth patterns from fractal patterns with D<2. We explain that Laplacian growth lies in the region belonging to two-dimensional growth patterns, motivating the main conjecture of this paper, i.e., that Laplacian growth patterns are two 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 ultraviolet cutoff length.

Algorithm for parallel Laplacian growth by iterated conformal maps.

We report an algorithm to generate Laplacian growth patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth patterns are compared to those obtained by the best algorithms of direct numerical solutions. The fractal dimension of the patterns is discussed.