arXiv subjects

Benny Davidovitch

Publications and source records attributed to Benny Davidovitch.

40 records · Page 3Linked to original sources

arXiv2001

Thermodynamic Formalism of the Harmonic Measure of Diffusion Limited Aggregates: Phase Transition and Converged $f(α)$

We study the nature of the phase transition in the multifractal formalism of the harmonic measure of Diffusion Limited Aggregates (DLA). Contrary to previous work that relied on random walk simulations or ad-hoc models to estimate the low probability events of deep fjord penetration, we employ the method of iterated conformal maps to obtain an accurate computation of the probability of the rarest events. We resolve probabilities as small as $10^{-70}$. We show that the generalized dimensions $D_q$ are infinite for $q<q^*$, where $q^*= -0.17\pm 0.02$. In the language of $f(α)$ this means that $α_{max}$ is finite. We present a converged $f(α)$ curve.

cond-mat.stat-mech↗

arXiv2001

Iterated Conformal Dynamics and Laplacian Growth

The method of iterated conformal maps for the study of Diffusion Limited Aggregates (DLA) is generalized to the study of Laplacian Growth Patterns and related processes. We emphasize the fundamental difference between these processes: DLA is grown serially with constant size particles, while Laplacian patterns are grown by advancing each boundary point in parallel, proportionally to the gradient of the Laplacian field. We introduce a 2-parameter family of growth patterns that interpolates between DLA and a discrete version of Laplacian growth. The ultraviolet putative finite-time singularities are regularized here by a minimal tip size, equivalently for all the models in this family. With this we stress that the difference between DLA and Laplacian growth is NOT in the manner of ultraviolet regularization, but rather in their deeply different growth rules. The fractal dimensions of the asymptotic patterns depend continuously on the two parameters of the family, giving rise to a "phase diagram" in which DLA and discretized Laplacian growth are at the extreme ends. In particular we show that the fractal dimension of Laplacian growth patterns is much higher than the fractal dimension of DLA, with the possibility of dimension 2 for the former not excluded.

cond-mat.stat-mech↗

arXiv2001

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 1-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.

cond-mat.stat-mech↗

arXiv2000

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 \to \infty) clusters. The computed dimension is D=1.713\pm 0.003.

cond-mat.stat-mech↗