Numerical simulations of diffusion - limited aggregation on the torus

Abstract

The results of numerical simulations of diffusion-limited aggregation on the torus are presented. The usual random walk was generalised by allowing the particle to perform jumps of length equal to s lattice spacings, s 3 1. Patterns with periodic structure were obtained. Recently there has been an increasing interest in the study of irreversible kinetic processes leading to the formation of fractal patterns. A simple stochastic model for the formation of clusters of particles in two-dimensional space was proposed by Witten and Sander (1981, 1983). In their model, called diffusion-limited aggregation (DLA), a single particle walks randomly on the square lattice until it reaches another particle (‘seed’), located usually in the centre of the lattice. Next, a new particle initiates its random walk. If the particle contacts the cluster (now built of two particles) it is incorporated into the cluster and the cluster grows. This process is repeated many times and leads to ramified structures possessing remarkable scaling properties (see figure 1, where a cluster of 3000 particles is shown). For example, the number N of Figure 1. Typical aggregate obtained from 3000 particles by means of usual DLA on the torus. 0305-4470/88/030159 + 06$02.50 @ 1988 IOP Publishing Ltd L159 L160 Letter to the Editor particles contained inside the circle of radius R grows as N ( R ) R D (1) where D = 1.7 is the fractal dimension (Mandelbrot 1982). It is believed that the value 1.7 for D is universal, i.e. it does not depend on the lattice, which was confirmed for the triangular lattice by Witten and Sander (1981, 1983) and for the non-lattice case by Meakin (1983a). Some modifications and different techniques for computer simulations have been discussed (for reviews see Sander (1985), Stanley and Ostrovsky (1986) and Herrmann (1986)). Variants of the original DLA include the introduction of the probability distribution for the sticking of particles (Meakin 1983a), superimposition of the drift on the diffusion (Meakin 1983b), using so-called Levy flight instead of the usual diffusion (Meakin 1984), modification of the sticking rule (Kertksz and Vicsek 1986), and so on. Also, electric breakdowns in dielectrics lead to fractal structures for the discharge patterns (Niemeyer et a1 1984). Some different algorithms for computer simulations have also been used (see, e.g., Meakin (1983a) and Ball and Brady (1985)). The aim of these modifications was the reduction of the duration of computer simulations. For example, in Meakin’s (1983a) simulations a particle was killed if it went sufficiently far away from the cluster and the new particle started on the circle surrounding the aggregate. To our knowledge, there is still no satisfactory theory of DLA (see, however, Gould et a1 (1983), Muthukumar (1983) and Halsey er a1 (1986)). In this letter we shall present preliminary results of the computer simulation of DLA on the torus. The recipe for the formation of aggregates was the following. Let us imagine the square lattice with L sites along one edge. One of the edges was chosen with probability a and on it the starting point for the random walk was chosen with probability 1/L. We generalised the usual random walk by allowing the particle to perform jumps of length equal to s lattice spacings, s 2 1, with probability a in one of four directions: up, down, left and right. We imposed on the random walk the periodic boundary condition that a particle crossing one of the edges appears on the opposite side (so the number of bonds between sites is also equal to L ) . In other words, the particle performs a random walk on the torus. As we took the number L to be prime (see below) it was possible to locate the seed in the centre of the square. The sticking rule was the usual one, i.e. the particle was incorporated into the aggregate on the first contact with it, providing that one particle only can occupy each site. (For s = 1, i.e. for usual DLA, it is impossible that the walker will fall into the site already occupied.) We should add that we allowed the particle to walk on the sites already occupied, i.e. on the aggregate. For the termination of the aggregation process, the trajectory of each particle should reach the cluster. In other words, the trajectory should fill out the whole torus. It is obvious that for usual diffusion with s = 1 each site can be reached by the walker. But in a case of the random walk with length of step s > 1, the trajectory will fill out the torus only when L and s are mutually prime, i.e. when the greatest common divisor (GCD) of L and .s is equal to 1. It is common to denote the GCD of two natural numbers a and b by (a, b ) , so for the ‘ergodicity’ of the random walk the following condition should be satisfied: ( L , s) = 1. (2) The above condition can be justified in the following way. Using the following identity: ( A mod C + B ) mod C = ( A + B ) mod C Letter to the Editor L161 the x coordinate, for example, of the particle after some number of jumps can be written as x = (xo+ k s ) mod L = xo+ ks L[(xo+ k s ) / L ] k E Z where [ r ] denotes the integer part of the number r and xo is a starting point. Denoting q = [(xo+ ks ) / L ] and shifting x xo+ x we can write

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Cite this paper

@inproceedings{Wolf1987NumericalSO, title={Numerical simulations of diffusion - limited aggregation on the torus}, author={Marek Wolf}, year={1987} }