14 research outputs found
Diffusion Limited Aggregation with Power-Law Pinning
Using stochastic conformal mapping techniques we study the patterns emerging
from Laplacian growth with a power-law decaying threshold for growth
(where is the radius of the particle cluster). For
the growth pattern is in the same universality class as diffusion
limited aggregation (DLA) growth, while for the resulting patterns
have a lower fractal dimension than a DLA cluster due to the
enhancement of growth at the hot tips of the developing pattern. Our results
indicate that a pinning transition occurs at , significantly
smaller than might be expected from the lower bound
of multifractal spectrum of DLA. This limiting case shows that the most
singular tips in the pruned cluster now correspond to those expected for a
purely one-dimensional line. Using multifractal analysis, analytic expressions
are established for both close to the breakdown of DLA universality
class, i.e., , and close to the pinning transition, i.e.,
.Comment: 5 pages, e figures, submitted to Phys. Rev.
Elastic Scattering by Deterministic and Random Fractals: Self-Affinity of the Diffraction Spectrum
The diffraction spectrum of coherent waves scattered from fractal supports is
calculated exactly. The fractals considered are of the class generated
iteratively by successive dilations and translations, and include
generalizations of the Cantor set and Sierpinski carpet as special cases. Also
randomized versions of these fractals are treated. The general result is that
the diffraction intensities obey a strict recursion relation, and become
self-affine in the limit of large iteration number, with a self-affinity
exponent related directly to the fractal dimension of the scattering object.
Applications include neutron scattering, x-rays, optical diffraction, magnetic
resonance imaging, electron diffraction, and He scattering, which all display
the same universal scaling.Comment: 20 pages, 11 figures. Phys. Rev. E, in press. More info available at
http://www.fh.huji.ac.il/~dani
Universality in Bacterial Colonies
The emergent spatial patterns generated by growing bacterial colonies have
been the focus of intense study in physics during the last twenty years. Both
experimental and theoretical investigations have made possible a clear
qualitative picture of the different structures that such colonies can exhibit,
depending on the medium on which they are growing. However, there are
relatively few quantitative descriptions of these patterns. In this paper, we
use a mechanistically detailed simulation framework to measure the scaling
exponents associated with the advancing fronts of bacterial colonies on hard
agar substrata, aiming to discern the universality class to which the system
belongs. We show that the universal behavior exhibited by the colonies can be
much richer than previously reported, and we propose the possibility of up to
four different sub-phases within the medium-to-high nutrient concentration
regime. We hypothesize that the quenched disorder that characterizes one of
these sub-phases is an emergent property of the growth and division of bacteria
competing for limited space and nutrients.Comment: 12 pages, 5 figure
Small-particle limits in a regularized Laplacian random growth model
We study a regularized version of Hastings-Levitov planar random growth that models clusters formed by the aggregation of diffusing particles. In this model, the growing clusters are defined in terms of iterated slit maps whose capacities are given by c_n=c|\Phi_{n-1}'(e^{\sigma+i\theta_n})|^{-\alpha}, \alpha \geq 0, where c>0 is the capacity of the first particle, {\Phi_n}_n are the composed conformal maps defining the clusters of the evolution, {\theta_n}_n are independent uniform angles determining the positions at which particles are attached, and \sigma>0 is a regularization parameter which we take to depend on c. We prove that under an appropriate rescaling of time, in the limit as c converges to 0, the clusters converge to growing disks with deterministic capacities, provided that \sigma does not converge to 0 too fast. We then establish scaling limits for the harmonic measure flow, showing that by letting \alpha tend to 0 at different rates it converges to either the Brownian web on the circle, a stopped version of the Brownian web on the circle, or the identity map. As the harmonic measure flow is closely related to the internal branching structure within the cluster, the above three cases intuitively correspond to the number of infinite branches in the model being either 1, a random number whose distribution we obtain, or unbounded, in the limit as c converges to 0. We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis