Off-lattice Diffusion-limited Aggregation Research Articles

Dimension of diffusion-limited aggregates grown on a line.

Diffusion-limited aggregation (DLA) has served for 40 years as a paradigmatic example for the creation of fractal growth patterns. In spite of thousands of references, no exact result for the fractal dimension D of DLA is known. In this Letter we announce an exact result for off-lattice DLA grown on a line embedded in the plane D=3/2. The result relies on representing DLA with iterated conformal maps, allowing one to prove self-affinity, a proper scaling limit, and a well-defined fractal dimension. Mathematical proofs of the main results are available in N. Berger, E. B. Procaccia, and A. Turner, Growth of stationary Hastings-Levitov, arXiv:2008.05792.

Diffusion-Limited Aggregation with Polygon Particles

Diffusion-limited aggregation (DLA) assumes that particles perform pure random walk at a finite temperature and aggregate when they come close enough and stick together. Although it is well known that DLA in two dimensions results in a ramified fractal structure, how the particle shape influences the formed morphology is still unclear. In this work, we perform the off-lattice two-dimensional DLA simulations with different particle shapes of triangle, quadrangle, pentagon, hexagon, and octagon, respectively, and compare with the results for circular particles. Our results indicate that different particle shapes only change the local structure, but have no effects on the global structure of the formed fractal cluster. The local compactness decreases as the number of polygon edges increases.

Mode Selection of Diffusion-limited Aggregation

The diffusion-limited aggregation (DLA) is a pattern formation governed only by the nature of diffusion field. Irreversible sticking process of particles gives rise to isotropic random fractal structure of DLA. Since Witten and Sander proposed such a simulation model, many researchers have been devoted to various structures of offlattice DLA, such as self-similarity, active zone, multifractality, branching distribution, fractal dimension, and so on. However, many authors have indicated complex structures of DLA cluster. Mysterious one among them is near pentagonal symmetry with the outline shape of DLA cluster. A wedge model was proposed from the viewpoint of harmonic measure, which gives the fractal dimension D 1:714 for a symmetric cone of pentagon. The details of mysterious symmetry in DLA cluster are studied by numerical simulations. In order to study the exterior shape of DLA, we measure the coordinates of 10 sticking particles without cluster growth that is the same manner as the study of active zone. The sticking simulations are performed on the 2500 offlattice clusters composed of 3 10 particles. Here, simulation methods with high accuracy are used as reported in ref. 14. Averaged power spectrum of sticking distribution in angle direction is shown in Fig. 1 as a function of spectrum mode. The peak position of mean spectrum can be estimated as the mode of 5:4 by the fitting of parabolic curve near the peak. Thus, the mysterious symmetry of DLA is characterized by the near pentagon as indicated before reports. However, this peak mode is slightly increasing with increasing of cluster size in the present simulation up to 3 10 particles. Above ordinary DLA simulation is started from the initial condition of a seed particle with the radius RS 1⁄4 0. In order to clarify the stability of near pentagonal symmetry, following DLA simulations are performed on the initial conditions of circular seed up to RS 1⁄4 500. As shown in Fig. 2 for RS 1⁄4 500, it is obvious that the near pentagonal mode in angle direction is extremely stable for large size of cluster independent of initial condition. Cluster density as a function of angle is analyzed by power spectrum, and then spectrum peak is obtained from the mean data for 1000 clusters (30000 clusters for RS 1⁄4 0). An example of power spectrum along the cluster growth from the seed of RS 1⁄4 500 is drawn in Fig. 3(a), which shows decreasing of peak mode with increasing of cluster size. As shown in Fig. 3(b), the Fig. 1. Power spectrum of angle distribution for sticking particles onto the surface of 2500 ordinary DLA clusters composed of N 1⁄4 3000000 particles. The peak position is estimated as the mode of 5:4.

Off-lattice noise reduction and the ultimate scaling of diffusion-limited aggregation in two dimensions.

Off-lattice diffusion-limited aggregation (DLA) clusters grown with different levels of noise reduction are found to be consistent with a simple fractal fixed point. Cluster shapes and their ensemble variation exhibit a dominant slowest correction to scaling, and this also accounts for the apparent "multiscaling" in the DLA mass distribution. We interpret the correction to scaling in terms of renormalized noise. The limiting value of this variable is strikingly small and is dominated by fluctuations in cluster shape. Earlier claims of anomalous scaling in DLA were misled by the slow approach to this small fixed point value.

Tip-sticking probability and the branching distribution of a fractal pattern in a diffusion field

We study the number of m-branch subsets M( m) on a two-dimensional off-lattice diffusion-limited aggregation simulation, where m is the number of particles of an m-branch. The M( m) decays exponentially, however, for small m its behavior is not simple. The probability distribution of each length subset P( m)= M( m)/ M all is independent of cluster size, where M all = ∑ m M(m) . The mean branch length L ̄ ≈2.34 is found from the above distribution. The P( m) is represented by a branching dynamic model (BDM) including only S tip≈0.785 which is the tip-sticking probability of a Brownian particle. Moreover, L ̄ ≈2.34 is related to S tip≈0.785. Therefore, we understand that the branching distribution is decided only by the tip-sticking probability.

Statistical analysis of off-lattice diffusion-limited aggregates in channel and sector geometries.

Statistical analysis of off-lattice diffusion-limited aggregates in channel and sector geometries.

Local isotropy and geometry of particle flux lines in diffusion-limited aggregation

We present a theoretical description of the branch orientation and the geometry of walker flux lines in diffusion limited aggregation (DLA). It is based on the self-similarity and non-directedness of the branch structure and explains recent numerical results on the probability distribution of the branch orientation and the angle of particle attachment. Our results support the idea that asymptotically large off-lattice DLA clusters are locally isotropic and every individual walker flux line exhibits a certain type of self-similarity.

Number of branches in diffusion-limited aggregates: The skeleton.

We develop the skeleton algorithm to define the number of main branches $N_b$ of diffusion-limited aggregation (DLA) clusters. The skeleton algorithm provides a systematic way to remove dangling side branches of the DLA cluster and has successfully been applied to study the ramification properties of percolation. We study the skeleton of comparatively large ($\approx 10^6$ sites) off-lattice DLA clusters in two, three and four spatial dimensions. We find that initially with increasing distance from the cluster seed the number of branches increases in all dimensions. In two dimensions, the increase in the number of branches levels off at larger distances, indicating a fixed number of $N_b = 7.5\pm 1.5$ main branches of DLA. In contrast, in three and four dimensions, the skeleton continues to ramify strongly as one proceeds from the cluster center outward, and we find no indication of a constant number of main branches. Likewise, we find no indication for a fixed $N_b$ in a study of DLA on the Cayley tree. In two dimensions, we find strong corrections to scaling of logarithmic character, which can help to explain recently reported deviations from self-similar behavior.

Orientation of particle attachment and local isotropy in diffusion limited aggregation (DLA)

We simulate 50 off-lattice DLA clusters, one million particles each. The probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically. For increasing cluster size, $N$, the distribution crosses over extremely accurately to a cosine, whose amplitude decreases towards zero as a power-law in $N$. From this viewpoint, asymptotically large DLA clusters are locally $isotropic$. This contradicts previous conclusions drawn from density-density correlation measurements [P. Meakin, and T. Viscek, Phys. Rev. A {\bf 32}, 685 (1985)]. We present an intuitive phenomenological model random process for our numerical findings.

Comparative study of large-scale Laplacian growth patterns.

We investigate the scaling of cluster size with mass for our simulations of diffusion-limited aggregation (DLA) clusters and dielectric-breakdown (DB) clusters of ${10}^{6}$ particles grown on a square lattice, and DLA clusters of ${10}^{6}$ particles grown off lattice. We find that the mass distribution and scaling behavior for the on-lattice DB model are intermediate between those for the on-lattice DLA model and the off-lattice DLA model. We take this as evidence that signatures of lattice anisotropy and the microscopic kinetics of attachment are both manifest at large length scales.

The thermodynamics of fractals revisited with wavelets

The multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that very much like thermodynamic functions, the generalized fractal dimensions D q and the f( α) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal signals, the wavelets playing the role of “generalized boxes”. We illustrate our theoretical considerations on pedagogical examples, e.g., devil's staircases and fractional Brownian motions. We also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data. That we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects. In particular, we show that a dynamical system which leaves invariant such an object can be uncovered form the space-scale arrangement of its wavelet transform modulus maxima. We elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem. This step towards a statistical mechanics of fractals is illustrated on discrete period-doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos. Finally, we apply our technique to analyze the fractal hierarchy of DLA azimuthal Cantor sets defined by intersecting the inner frozen region of large mass off-lattice diffusion-limited aggregates (DLA) wit a circle. This study clearly lets out the existence of an underlying multiplicative process that is likely to account for the Fibonacci structural ordering recently discovered in the apparently disordered arborescent DLA morphology.

Multiscaling analysis and width of the active zone of large off-lattice DLA

We measure the multiscaling behavior of large off-lattice diffusion limited aggregates (DLA). In contrast to previous studies we now find a continuous dependence of the multiscaling dimensions D( x) on the relative distance x = r/ R g to the center of the cluster. This result agrees with measurements on smaller clusters. Furthermore we report the multiscaling behavior and the behavior of the width of the active zone of one very large off-lattice DLA cluster with 50 million particles. Here we find a sharp drop of the multiscaling dimensions for large x instead of a continuous behavior.

Multifractal scaling of 3D diffusion-limited aggregation

We study the multifractal (MF) properties of the set of growth probabilities { p i} for 3D off-lattice diffusion-limited aggregation (DLA). We find that: (i) the { p i} display MF scaling for all moments-in contrast to 2D DLA, where one observes a “phase transition” in the MF spectrum for negative moments; (ii) multifractality is also displayed by the p i located in a shell of reduced radius x ≡ r R g , where R g is the radius of gyration of the cluster and r the radius of the shell; (iii) the average value α av of α ≡ -In p/In M in a shell of reduced radius x in a cluster of mass M is a function that does not depend on the cluster mass but only on x.

Multifractal spectrum of off-lattice three-dimensional diffusion-limited aggregation

We study the multifractal properties of the set of growth probabilities {${\mathit{p}}_{\mathit{i}}$} for three-dimensional (3D) off-lattice diffusion-limited aggregation (DLA) in two distinct ways: (i) from the histogram of the ${\mathit{p}}_{\mathit{i}}$ and (ii) by Legendre transform of the moments of the distribution of ${\mathit{p}}_{\mathit{i}}$. We calculate the {${\mathit{p}}_{\mathit{i}}$} for 50 off-lattice clusters with cluster masses up to 15 000. We discover that for 3D DLA, in contrast to 2D DLA, there appears to be no phase transition in the multifractal spectrum. We interpret this difference in terms of the topological differences between two and three dimensions.

Structural five-fold symmetry in the fractal morphology of diffusion-limited aggregates

The statistical self-similarity of the geometry of diffusion-limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform. This study reveals the existence of a predominant structural five-fold symmetry in the internal frozen region as well as in the active outer region of the interface. This observation is corroborated by a statistical analysis of the screening effects that govern diffusion-limited aggregation (DLA) growth in linear and sector-shaped cells. The existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering. We report on the discovery of Fibonacci sequences in the inner extinct region of large mass off-lattice DLA clusters, with a branching ratio which converges asymptotically to the golden mean. We suggest an interpretation of the DLA morphology as a “quasifractal” counterpart of the well-ordered snowflake fractal architecture.

Branch order and ramification analysis of large diffusion-limited-aggregation clusters.

Using a hierarchical ordering scheme, many off-lattice diffusion-limited-aggregation (DLA) clusters containing ${10}^{6}$ particles are separated into branches of different orders. For each order we measure the number, mass, length, and width of the branches. All of these branch properties depend, with an exponential law, on the branch order. This means that for all of them the ratios between properties of subsequent branch orders are constant. By relating length, width, and mass to each other we find that the length and width of the whole branches both depend, with the same exponents ${\ensuremath{\nu}}_{\mathrm{?}}$=${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=0.60\ensuremath{\simeq}\ensuremath{\beta}=1/D, on the mass of the branch. By measuring the ramification matrix ${\mathit{R}}_{\mathit{i}\mathit{k}}$ of the clusters we find that the branches and stems are distributed in a self-similar way. Furthermore, we measure the angles enclosed by two branches. We find that this angle saturates to a finite value around 38\ifmmode^\circ\else\textdegree\fi{}. All of these results indicate a statistical self-similarity between branches of different orders. This result is supported by a direct comparison of off-lattice DLA clusters of ${10}^{5}$, ${10}^{6}$, and ${10}^{7}$ particles.

Multiscaling analysis of large-scale off-lattice DLA

An improved algorithm has been used to simulate off-lattice diffusion-limited aggregation (DLA) in two dimensions. This enables us to grow clusters of up to 6 × 10 6 particles, which are the largest such clusters made up to now. The density profile in two dimensions is analyzed for the scaling form (multiscaling) g( r, R) = r − d + D( x) C( x) with x = r R , where R is the radius of gyration and r is the distance from the cluster origin. In contrast with previous studies, the dimension D( x) shows no symmetric tendency to drop at values of x but only fluctuates around the global fractal dimension D = 1.712 ± 0.003.

Distribution of growth probabilities for off-lattice diffusion-limited aggregation.

We study the distribution n(\ensuremath{\alpha},M) of growth probabilities {${\mathit{p}}_{\mathit{i}}$} for off-lattice diffusion-limited aggregation (DLA) for cluster sizes up to mass M=20 000, where ${\mathrm{\ensuremath{\alpha}}}_{\mathit{i}}$==-${\mathit{p}}_{\mathit{i}}$/logM. We find that for large \ensuremath{\alpha}, log n(\ensuremath{\alpha},M)\ensuremath{\propto}-${\mathrm{\ensuremath{\alpha}}}^{\ensuremath{\gamma}}$/${\mathrm{log}}^{\mathrm{\ensuremath{\delta}}}$M, with \ensuremath{\gamma}=2\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\delta}=1.3\ifmmode\pm\else\textpm\fi{}0.3. One consequence of this form is that the minimum growth probability ${\mathit{p}}_{\mathrm{min}}$(M) obeys the asymptotic relation log${\mathit{p}}_{\mathrm{min}}$(M)\ensuremath{\sim}-(logM${)}^{(\ensuremath{\gamma}+1+\mathrm{\ensuremath{\delta}})/\ensuremath{\gamma}}$. We find evidence for the existence of a well-defined crossover value ${\mathrm{\ensuremath{\alpha}}}^{\mathrm{*}}$ such that only the rare configurations of DLA contribute to n(\ensuremath{\alpha},M) for \ensuremath{\alpha}>${\mathrm{\ensuremath{\alpha}}}^{\mathrm{*}}$, while both rare and typical DLA configurations contribute for \ensuremath{\alpha}${\mathrm{\ensuremath{\alpha}}}^{\mathrm{*}}$.

Diffusion-limited-aggregation-like displacement structures in a three-dimensional porous medium.

We have studied the structure that arises when a low-viscosity fluid displaces a high-viscosity fluid in a transparent three-dimensional porous medium. The viscosity ratio m is not far from 1 (m\ensuremath{\simeq}14) in this system. The fluids we use have equal densities so that there are no gravity effects. The injected fluid forms a ramified structure when an intermediate displacement rate is used (${\mathit{N}}_{\mathrm{Ca}}$\ensuremath{\simeq}${10}^{\mathrm{\ensuremath{-}}4}$). We have measured the projected area A of these growing structures. The cluster mass M is found to scale with a characteristic length ${\mathit{R}}_{\mathit{A}}$ obtained from this area, as M\ensuremath{\sim}${\mathit{R}}_{\mathit{A}\mathit{A}}^{\mathit{D}}$. We find the exponent to be ${\mathit{D}}_{\mathit{A}}$=2.5\ifmmode\pm\else\textpm\fi{}0.1. The physical model was simulated on the computer using the three-dimensional off-lattice diffusion-limited aggregation (DLA) algorithm. Projections of these clusters show features in common with the experimentally generated patterns. Quantitatively we find reasonably good agreement in terms of the exponent ${\mathit{D}}_{\mathit{A}}$, although the experiments are outside the DLA regime of viscous fingering.

Diffusion-limited aggregation

Improved algorithms have been developed for both off-lattice and hypercubic lattice diffusion-limited aggregation (DLA) in dimensionalities ( d ) 3-8 and for two-dimensional off-lattice DLA. In two-dimensional off-lattice DLA a fractal dimensionality ( D ) of about 1.71 was obtained for clusters containing up to 10 6 particles. This is significantly larger than the value of ( d 2 + 1)/( d + 1) (5/3 for d = 2) predicted by mean field theories. For d > 2 the off-lattice simulations give results that are consistent with the mean field theories. For d = 3 and d = 4 the effects of lattice anisotropy can easily be seen for clusters containing 3 x 10 6 and 10 6 sites respectively and the effective fractal dimensionalities are slightly smaller for the lattice model clusters than for the off-lattice clusters. Results are also presented for two-, three- and four- dimensional lattice model clusters with noise reduction.