Scaling in a ballistic aggregation model

Liang, Shoudan; Kadanoff, Leo P.

doi:10.1103/physreva.31.2628

Cited by 42 publications

(15 citation statements)

References 9 publications

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“…This scaling hypothesis was also used by Liang and Kadanoff [10] to study the driven ballistic aggregation, in which the particles trajectories are in a single direction. They conclude that the γ exponent is nonuniversal, i.e., depends on the lattice structure.…”

Section: Resultsmentioning

confidence: 99%

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Morphological transition between diffusion-limited and ballistic aggregation growth patterns

et al. 2005

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In this work, the transition between diffusion-limited and ballistic aggregation models was revisited using a model in which biased random walks simulate the particle trajectories. The bias is controlled by a parameter λ, which assumes the value λ = 0 (1) for ballistic (diffusion-limited) aggregation model. Patterns growing from a single seed were considered. In order to simulate large clusters, a new efficient algorithm was developed. For λ = 0, the patterns are fractal on the small length scales, but homogeneous on the large ones. We evaluated the mean density of particles ρ in the region defined by a circle of radius r centered at the initial seed. As a function of r, ρ reaches the asymptotic value ρ0(λ) following a power law ρ = ρ0 + Ar −γ with a universal exponent γ = 0.46(2), independent of λ. The asymptotic value has the behavior ρ0 ∼ |1 − λ| β , where β = 0.26(1). The characteristic crossover length that determines the transition from DLA-to BA-like scaling regimes is given by ξ ∼ |1 − λ| −ν , where ν = 0.61(1), while the cluster mass at the crossover follows a power law M ξ ∼ |1 − λ| −α , where α = 0.97(2). We deduce the scaling relations β = νγ and β = 2ν − α between these exponents.

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Section: Resultsmentioning

confidence: 99%

“…If the random walks are replaced by ballistic trajectories at random directions, we have the ballistic aggregation (BA) model [9]. In contrast to DLA, the BA model generates disordered nonfractal clusters with nontrivial scaling properties [2,10].…”

Section: Introductionmentioning

confidence: 99%

Morphological transition between diffusion-limited and ballistic aggregation growth patterns

et al. 2005

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“…Although progress has been made with the Eden model (74)(75)(76) and in the ballistic case (63,77), results have been unsatisfactory in the case of DLA (and related processes) where growth is controlled by diffusing particles or fields. Here we lack even a schematic model to show why this type of growth leads to a scale-invariant structure.…”

Section: Discussionmentioning

confidence: 96%

Tenuous Structures from Disorderly Growth Processes

Witten

Cates

1986

Science

133

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Colloidal aggregation and other random growth processes produce structures that behave differently from ordinary bulk matter. Much of this behavior can be described in terms of the invariance of the aggregates under changes of spatial length scale: they appear to be fractals. There are two types of basic mechanisms for producing fractal aggregates. Those in which aggregation proceeds cluster by cluster can be understood qualitatively in terms of a solvable schematic model. The diffusion-limited aggregation or deposition of individual particles to make a large cluster is not as well understood. It is closely related to several irreversible processes in other areas of physics, such as two-fluid displacement in porous materials and the dielectric breakdown of insulators. More generally, disorderly growth mechanisms provide structures having unique properties, many of which can be understood by using simple statistical principles.

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“…It was recognized that many important kinetic aggregation processes result in fractal geometry, such as electron deposition and viscous fingering [1]. To study the mechanism of fractal pattern formation, several models have been considered in recent years, such as the diffusion-limited aggregation (DLA) model [2] and the ballistic aggregation model [3]. In particular, an irreversible kinetic aggregation lattice model called the shortest-path aggregation (SPA) has been introduced [4].…”

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Kinetic real-space renormalization-group approach to the shortest-path aggregation

Wang

1994

Phys. Rev. E

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A kinetic real-space renormalization-group approach to the shortest-path aggregation (SPA) is presented. The fractal dimension (Df) and a set of hierarchical dimensions D(q) are computed for the SPA cluster on the square lattice. In particular, 2X2 and 3 X 3 cell real-space renormalization groups have been carried out. We find Df = Fig. 1.In the lattice-bond version of the SPA, using the KRG approach of Wang et ai. , we thus distinguish between three types of bonds at each order n of the transformation: bulk bonds representing the particles on the cluster with mass M"(bold lines in Fig. 1); perimeter bonds with mass m"(wavy lines), representing the surface of the aggregate on which the next potential growth may occur; and massless empty bonds for the rest (thin lines).A cluster grows by changing perimeter bonds to bulk bonds one by one according to probabilities which are determined by the growth mechanism. As a particle is added to the cluster one perimeter bond turns into a bulk bond and all its unoccupied neighbors become growth bonds. The following rules guide the transformation: If the cell is connected from top to bottom by bulk bonds it will be renormalized into a bulk bond; if the cell is completely empty it will be renormalized to an empty bond; all other configurations are renormalized into a perimeter bond. Therefore all configurations in Fig. 1(a) will be renormalized to the vertical perimeter bonds, and all configurations in Fig. 1(b) will be renormalized to vertical bulk bonds.In order to find the local growth probabilities on each bond, we notice that the particles are released from the top and adhere to perimeter bonds along the shortest path, and, if there is more than one such path for a particular particle, then each path has an equal chance to be chosen. Let us take configuration 1 of Fig. 1(a) an an example. If we denote p, , and p, 2 as the growth probabilities on the perimeter bonds 1 and 2, respectively. Ac-

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Scaling in a ballistic aggregation model

Cited by 42 publications

References 9 publications

Morphological transition between diffusion-limited and ballistic aggregation growth patterns

Morphological transition between diffusion-limited and ballistic aggregation growth patterns

Tenuous Structures from Disorderly Growth Processes

Kinetic real-space renormalization-group approach to the shortest-path aggregation

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