Fereydoon Family

Dynamics of fractal surfaces

Self-affine fractal geometry, B.B. Mandelbrot, et al dynamic scaling, J.M. Hammersley, et al models - models of growing rough surfaces - ballistic deposition, M.J. Vold, et al the eden growth model, M. Eden, et al continuum equations and directed polymers, M. Kardar, et al other surface growth models, Y.C. Zhang, et al experiments - surface growth experiments, C. Licoppe, et al bibliography index.

Kinetics of aggregation and gelation

We define six kinetic growth models that have witnessed an explosion of recent activity: 1 Cancer Growth (Eden) 2 Colloid Growth (Langer-Muller-Krumbhaar; Witten-Sander) 3 Breakdown (Sawada et al.) 4 Crystallization (Rikvold) 5 Invasion Percolation (Schlumberger group) 6 Addition Polymerization (Manneville-de Seze; Herrmann et al.) We also describe briefly some of the approaches used to study these models, with emphasis on the re normalization group approach being developed by us.

Dynamic scaling and phase transitions in interface growth

The dynamic scaling approach is an effective tool for understanding the temporal evolution of fluctuating interfaces. The surface width w obeys the scaling form w(L, t)=Lαƒ(t/Lαβ, where α and β are exponents which characterize how the surface width grows with the length scale L and the time t. Applications of dynamic scaling to a number of surface growth models are discussed. The results of a large-scale simulation of the ballistic deposition model in three dimensions are presented and compared with recent conjectures. The question of universality in interface growth is addressed and a finite temperature generalization of the restricted solid-on-solid model is studied. This model appears to undergo a phase transition from the usual rough phase at high temperatures to a new phase at low temperatures. Numerical solutions of the generalized Langevin equation indicate that there is no phase transition in 2+1 dimensions. Thus, the relation between the continuum equation and the discrete models is still an open question.

Fractal pattern formation in human retinal vessels

Abstract The mechanism for the formation of retinal vessel patterns in the developing human eye is an unresolved question of considerable importance. The current hypothesis is based on the existence of a variable oxygen gradient across the developing photoreceptors which stimulates the release of angiogenic factors which diffuse in the plane of the retina and result in the growth of retinal vessels. This implies that the limiting step in the formation of retinal blood vessels is a diffusion process. To test this hypothesis we have performed a fractal analysis of the human retinal vessels using two different methods. Within the limited range of length scales available in the red-free fundus photographs, we find that the human retinal blood vessels have a self-similar structure with a fractal dimension D ≈ 1.7. Since this value of D is the same as the value found for a diffusion limited growth process, our result supports the hypothesis that diffusion is the fundamental process in the formation of human retinal vessel patterns.

Active random walkers simulate trunk trail formation by ants

A simple model for interactive structure formation is studied to simulate the trail formation by ants based on local chemical communication. In our model, active random walkers, which do not have the ability of visual navigation or storage of information, first have to discover different distributions of food sources and then have to link these sources to a central place by forming a trail, using no other guidance than the chemical markings produced by themselves. The simulations show the spontaneous emergence of a collective trail system due to self-organization, which is both stable and flexible, to include newly discovered sources. The typical dendritic foraging patterns of desert ants, reported by Hölldobler and Möglich (Insectes Sociaux. 1980. 27(3). pp. 237 264) are reproduced by the simulations.

Cluster size distribution in chemically controlled cluster–cluster aggregation

The dynamics of the cluster–cluster aggregation process is investigated through the time dependent cluster size distribution function using Monte Carlo simulations, scaling theory, and the Smoluchowski coagulation equation. Depending on such factors as the chemical reactivity, kinetic energy, mass, etc. of the aggregates the coagulation of two clusters may or may not take place. These effects are simulated by assuming that the probability that two clusters of sizes i and j irreversibly stick together is proportional to (ij)σ. Our results show that for constant small sticking probability cluster size distribution and its moments asymptotically scale with the same exponents as for the case when the sticking probability is unity. In the early stages, coagulation is slow and the process is chemically controlled. However, for finite sticking probability, as the aggregation process develops in time the chance that two clusters join permanently increases with the surface of the cluster and there is a crossover f...

Multifractal Geometry of Diffusion-Limited Aggregates

Off-lattice diffusion-limited aggregation (DLA) clusters are shown to have multifractal geometry. We determine the Dq spectrum associated with the mass distribution using an improved sand box technique and demonstrate that they decrease monotonically with increasing q. From our results it follows that the commonly used methods for the determination of the fractal dimension of DLA clusters yield Dq=2 which is smaller than the true fractal dimension Dq=0.

Dynamic scaling of the interface in two-phase viscous flows in porous media

The time evolution and geometry of rough interfaces observed in experiments on immiscible displacement of viscous fluids in porous media are analysed using the concepts of dynamic scaling and self-affine fractal geometry. The authors find that the development of the interface is governed by dynamic scaling and they have determined the corresponding surface exponents alpha and beta . The values of the exponents calculated for the experimental patterns are different from those obtained for a variety of two-dimensional models of marginally stable interfaces.

Fractal Dimension and Grand Universality of Critical Phenomena

Conformation of branched random fractals formed in equilibrium processes is discussed using a Flory-type theory. Within this approach we find only three distinct types or classes of random fractals. We call these theextended, thecompensated, and thecollapsed states. In particular, the critical clusters in thermal phase transitions are found to be of the compensated type and have approximately the same value of the fractal dimension. The Flory theory predicts the upper critical dimension for these clusters to be 6 instead of 4. This result and the apparent “grand” universality of the fractal geometry of the clusters in critical phenomena are discussed.

Adhesion failures determine the pattern of choroidal neovascularization in the eye: a computer simulation study.

Choroidal neovascularization (CNV) of the macular area of the retina is the major cause of severe vision loss in adults. In CNV, after choriocapillaries initially penetrate Bruchs membrane (BrM), invading vessels may regress or expand (CNV initiation). Next, during Early and Late CNV, the expanding vasculature usually spreads in one of three distinct patterns: in a layer between BrM and the retinal pigment epithelium (sub-RPE or Type 1 CNV), in a layer between the RPE and the photoreceptors (sub-retinal or Type 2 CNV) or in both loci simultaneously (combined pattern or Type 3 CNV). While most studies hypothesize that CNV primarily results from growth-factor effects or holes in BrM, our three-dimensional simulations of multi-cell model of the normal and pathological maculae recapitulate the three growth patterns, under the hypothesis that CNV results from combinations of impairment of: 1) RPE-RPE epithelial junctional adhesion, 2) Adhesion of the RPE basement membrane complex to BrM (RPE-BrM adhesion), and 3) Adhesion of the RPE to the photoreceptor outer segments (RPE-POS adhesion). Our key findings are that when an endothelial tip cell penetrates BrM: 1) RPE with normal epithelial junctions, basal attachment to BrM and apical attachment to POS resists CNV. 2) Small holes in BrM do not, by themselves, initiate CNV. 3) RPE with normal epithelial junctions and normal apical RPE-POS adhesion, but weak adhesion to BrM (e.g. due to lipid accumulation in BrM) results in Early sub-RPE CNV. 4) Normal adhesion of RBaM to BrM, but reduced apical RPE-POS or epithelial RPE-RPE adhesion (e.g. due to inflammation) results in Early sub-retinal CNV. 5) Simultaneous reduction in RPE-RPE epithelial binding and RPE-BrM adhesion results in either sub-RPE or sub-retinal CNV which often progresses to combined pattern CNV. These findings suggest that defects in adhesion dominate CNV initiation and progression.