Colored diffusion-limited aggregation for urban migration

In this paper we study the growth probability and cluster morphologies which emerge in an off-lattice, two-dimensional, colored diffusion-limited aggregation model for urban dynamics, particularly migration. To reach this goal, three immobile interacting clusters that include the geographical concept of gravity are studied by exact enumeration. In our simulations we find a strong correlation between the seed’s distance, migration rules and number of aggregated particles. The growth probability of a certain angular subset and its rate and route of convergence to a Normal distribution when migration cost is acting are also shown. We search how all the factors mentioned above determine the cluster morphologies.     

Angular gaps in radial diffusion-limited aggregation: two fractal dimensions and nontransient deviations from linear self-similarity

When suitably rescaled, the distribution of the angular gaps between branches of off-lattice radial diffusion-limited aggregation is shown to approach a size-independent limit. The power-law expected from an asymptotic fractal dimension D = 1.71 arises only for very small angular gaps, which occur only for clusters significantly larger than M = 10(6) particles. Intermediate size gaps exhibit an effective dimension around 1.67, even for M--> infinity. They dominate the distribution for clusters with M<10(6). The largest gap approaches a finite limit extremely slowly, with a correction of order M(-0.17).     

Cluster-cluster correlation during irreversible diffusion-limited aggregation

Summary In this paper, we discuss the evolution of the scattered intensityI(q) during irreversible diffusion-limited cluster-cluster aggregation. We analyse twodimensional simulations and interpret the results within the framework of a recently proposed theoretical approach. The theory describes the correlation among different clusters which develops during the irreversible aggregation process. The model is based on two coupled differential equations, controlling the growth of the average cluster mass and the time dependence of the probability of finding pairs of clusters as a function of their distance. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Thermodynamic and scaling behaviour in finite diffusion-limited aggregation

Rényi's entropies for diffusion-limited aggregates are studied as a function of the number N of particles contained in the aggregates. It is found that Rényi's values increase with log N in a linear fashion, and that the aggregates exhibit multifractal behaviour for finite values of N. When N → ∞, the aggregate has a monofractal structure. Rényi's entropies depend on the fractal dimension of the aggregate. When the fractal dimension increases, the values of K_q decrease for q ? 1> and increase for q > 1.     

Effects of the screening breakdown in the diffusion-limited aggregation model

Several models based on the diffusion-limited aggregation (DLA) model were proposed and their scaling properties explored by computational and theoretical approaches. In this paper, we consider a new extension of the on-lattice DLA model in which the unitary random steps are replaced by random flights of fixed length. This procedure reduces the screening for particle penetration present in the original DLA model and, consequently, generates new pattern classes. The patterns have DLA-like scaling properties at small length of the random flights. However, as the flight size increases, the patterns are initially round and compact but become fractal for sufficiently large clusters. Their radius of gyration and number of particles at the cluster surface scale asymptotically as in the original DLA model. The transition between compact and fractal patterns is characterized by wavelength selection, and 1/k noise was observed far from the transition.Received: 2 March 2004, Published online: 14 December 2004PACS: 05.40.Fb Random walks and Levy flights - 05.50. + q Lattice theory and statistics (Ising, Potts, etc.) - 05.10.Ln Monte Carlo methods     

Fractal dimensions of time sequences

We present a simple and efficient way for calculating the fractal dimension $D$ of any time sequence sampled at a constant time interval. We calculated the error of a piecewise interpolation to $N+1$ points of the time sequence with respect to the next level of ($2N+1$)-point interpolation. This error was found to be proportional to the scale (i.e., $1/N$) to the power of $1?D$. A simple analysis showed that our method is equivalent to the inverse process of the method of random midpoint displacement widely used in generating fractal Brownian motion for a given $D$. The efficiency of our method makes the fractal dimension a practical tool in analyzing the abundant data in natural, economic, and social sciences.     

Fractal wavelet dimensions and localization

In this paper we want to give a new definition of fractal dimensions as small scale behavior of theq-energy of wavelet transforms. This is a generalization of previous multi-fractal approaches. With this particular definition we will show that the 2-dimension (=correlation dimension) of the spectral measure determines the long time behavior of the time evolution generated by a bounded self-adjoint operator acting in some Hilbert space ?. It will be proved that for φ, ψ∈? we have $$\mathop {\lim \inf }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ + (2)$$ and that $$\mathop {\lim \sup }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ - (2),$$ wherek ^±(2) are the upper and lower correlation dimensions of the spectral measure associated with ψ and ?. A quantitative version of the RAGE theorem shall also be given.     

Fractal dimensions and homeomorphic conjugacies

We investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes. When the homeomorphic conjugacy is not Lipschitz continuous, we discuss how its singularities can affect the whole set of generalized fractal dimensions. We give applications to homeomorphisms that conjugate critical circle maps with irrational (golden mean) winding numbers. We present numerical studies corroborating the theoretical predictions.     

Particle aggregation versus cluster aggregation in high dimensions

We distinguish two different types of irreversible aggregation-accretion of individual particles and successive aggregation of clusters of comparable size. In aggregation of particles which follow trajectories of fractal dimensionD ₁, we show that physical limits on the aggregation rate impose a lower bound on the fractal dimensionD ₀ of the aggregate. Ind-dimensional space,D ₀{d–D}₁ + 1. Thus aggregation of ballistic particles, withD ₁ = 1, is not fractal. By contrast, cluster aggregates appear to attain a finite, limitingD ₀ in high dimensions. We present a soluble model with this property, and argue that it should agree with Sutherland's binary aggregation model in high dimensions. For this model,D ₀ depends continuously on a parameter; the exponent is not universal.     

A new example of the diffusion-limited aggregation: Ni-Cu film patterns

The mechanism of the growth of the dendrites in the Ni-Cu films is studied by comparing them with the aggregates obtained by Monte Carlo (MC) simulations according to the diffusion-limited aggregation (DLA) model. The films were grown by electrodeposition. The structural analysis of the films carried out using the x-ray diffraction showed that the films have a face-centered cubic structure. Scanning electron microscope (SEM) was used for morphological observations and the film compositions were determined by energy dispersive x-ray spectroscopy. The observed SEM images are compared with the patterns obtained by MC simulations according to DLA model in which the sticking probability, P between the particles is used as a parameter. For all samples between the least and the densest aggregates in the films, the critical exponents of the density-density correlation functions, α were within the interval 0.160 ± 0.005-0.124 ± 0.006, and the fractal dimensions, D_f, varies from 1.825 ± 0.006 to 1.809 ± 0.008 according to the method of two-point correlation function. These values are also verified by the mass-radius method. The pattern with α and D_f within these intervals was obtained by MC simulations to DLA model while the sticking probability, P was within the interval from 0.35 to 0.40 obtained by varying P (1-0.001). The results showed that the DLA model in this binary system is a possible mechanism for the formation of the ramified pattern of Ni-Cu within the Ni-rich base part of the Ni-Cu films due to the diffusive characteristics of Cu.