How anisotropy beats fractality in two-dimensional on-lattice DLA growth
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug How anisotropy beats fractality in two-dimensional on-lattice DLA growth
Denis S. Grebenkov
1, 2, ∗ and Dmitry Beliaev † Laboratoire de Physique de la Mati`ere Condens´ee (UMR 7643),CNRS – Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau, France Interdisciplinary Scientific Center Poncelet (ISCP), ‡ Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia Mathematical Institute, University of Oxford,Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK (Dated: Received: August 7, 2017/ Revised version:)We study the fractal structure of Diffusion-Limited Aggregation (DLA) clusters on the squarelattice by extensive numerical simulations (with clusters having up to 10 particles). We observethat DLA clusters undergo strongly anisotropic growth, with the maximal growth rate along theaxes. The naive scaling limit of a DLA cluster by its diameter is thus deterministic and one-dimensional. At the same time, on all scales from the particle size to the size of the entire cluster ithas non-trivial box-counting fractal dimension which corresponds to the overall growth rate which,in turn, is smaller than the growth rate along the axes. This suggests that the fractal nature of thelattice DLA should be understood in terms of fluctuations around one-dimensional backbone of thecluster. PACS numbers: 05.40.-a, 05.50.+q, 05.65.+b, 05.10.LnKeywords: DLA, non-equilibrium growth, anisotropy, fractals
INTRODUCTION
Diffusion Limited Aggregation (DLA) was first intro-duced by Witten and Sander [1, 2] as a model of irre-versible colloidal aggregation and then rapidly became abasic model of non-equilibrium growth phenomena suchas electrodeposition and dendritic growth, viscous finger-ing in fluids, dielectric breakdown, mineral deposition,bacterial colony growth, pattern formation, to name buta few [3–11]. The growth is driven by a Laplacian fieldand is modeled by adding particles, one at a time, to agrowing cluster via either a random walk on a lattice, orBrownian motion. In spite of these very simple growthrules, only a few rigorous mathematical results aboutDLA are available [12, 13]. Most properties of both on-lattice and off-lattice DLA clusters are known either fromnumerical simulations, or from theoretical approxima-tions (see [14–21] and references therein). In particular,numerical simulations have revealed that DLA clusters onthe square lattice are inhomogeneous [22–24], anisotropic[17, 25–28] and multifractal [29, 30]; their properties arelattice dependent (i.e., nonuniversal) [27]; their scaling isnot determined by a single exponent [24, 31]; and the in-volved “exponents” change with the number of particlessuggesting a transient regime [24, 27]. To some extent, allthese properties are caused by the local anisotropy of thelattice growth rules. As a consequence, the scaling limitof the on-lattice DLA remains controversial. This situ- ‡ International Joint Research Unit – UMI 2615 CNRS/ IUM/ IITPRAS/ Steklov MI RAS/ Skoltech/ HSE, Moscow, Russian Federa-tion ation contrasts with the significant progress made overthe last decade in the analysis of other lattice modelssuch as percolation and Ising models. The identificationof stochastic Loewner evolution (SLE) processes as thescaling limit of lattice models led to numerous break-through discoveries in this field of statistical physics andmathematics [32–34].In this paper, we provide theoretical arguments andextensive numerical simulations to shed a light onto thescaling limit of on-lattice DLA clusters. Our main con-clusion is that the naive but widely used scaling limit, inwhich the cluster is rescaled by its diameter, is a deter-ministic one-dimensional cross-like shape. Figurativelyspeaking, anisotropy of the cluster beats fractality, re-sulting in a trivial, non-fractal limit. To explain thispoint, let us consider a graph of a one-dimensional ran-dom walk (with unit-size steps) versus the number ofsteps t . This is a random curve on the plane. Rescalingof this curve by its diameter (which is equal to t in thissetting) yields a trivial deterministic limit – the unit in-terval. In order to obtain a nontrivial limit (the Brownianpath), anisotropic rescaling has to be performed, by t and √ t , along the horizontal and vertical axes, respectively.While the choice of rescaling factors is elementary forthis toy example, the proper rescaling of an anisotropicon-lattice DLA cluster remains unknown.The scaling properties of DLA clusters are usuallycharacterized by two observables: the growth rate β andthe fractal dimension D . Most authors compute the lat-ter using the former. Indeed, if one covers a regular frac-tal of diameter 1 by disks of size ǫ , then the numberof disks scales as N ∝ ǫ − D , where D is the Minkowskidimension of the fractal. Rescaling the fractal by ǫ − yields the diameter ∝ N D . Hence the growth rate is the FIG. 1: A large DLA cluster with with 145 199 976 particles.In this coarse-grained picture of resolution 2048 × ×
64 block of the original cluster imageof size 2 × . inverse of the dimension. This relation that was first putforward by Stanley for percolation clusters [35] (see also[36]), was often used to get the fractal dimension of bothon-lattice and off-lattice DLA clusters by computing thegrowth rate for the radius of gyration (e.g., [27]). It isimportant to stress, however, that this relation does nothold in general, it is valid only under some regularityassumptions. The simplest counter-example is an aggre-gate of 2 t particles, half of them forming a disk of radius ∝ √ t , and the other half forming an interval of length ∝ t . For this aggregate the fractal dimension is 2 (deter-mined by the disk) but the growth rate is 1 (determinedby the interval). The naive rescaling by the diameter ∝ t results in a trivial limit (the unit interval) because thepart with the higher dimension but smaller growth rate(the disk) is shrunk and thus fully eliminated in the limit t → ∞ .To our knowledge, the above regularity assumptionand the consequent equality between the inverse of thegrowth rate β and the fractal dimension D were neverproperly verified for the on-lattice DLA. The first goal ofour work is to check this important equality. Althoughwe obtain slightly different numerical values for D and1 /β , they cannot be distinguished within the numericalaccuracy. The second goal consists in emphasizing therole of anisotropy. For this purpose, we introduce theangular growth rate and show that DLA clusters growfaster along the axes of the square lattice. In particularthis implies that if the DLA cluster is rescaled by its di-ameter, then the scaling limit becomes deterministic and one-dimensional. In other words, the parts of the DLAcluster with lower growth rates are eliminated, as in theabove example with a disk and an interval. One can in-terpret this result as a kind of the law of large numbersfor DLA clusters. On the other hand, branches of DLAexhibit a strong pre-fractal behavior that suggests thatfluctuations of DLA branches around the axes have non-trivial scaling limit. This observation can be interpretedas an analogue of the central limit theorem. NUMERICAL RESULTS
Our strategy to support the above claims consists intwo parts: (i) numerical computation of both the growthrate β and the fractal dimension D , and (ii) profoundanalysis of the cluster anisotropy. For this purpose, weadapted a bias-free algorithm by Y. E. Loh to generateDLA clusters on the square lattice [37]. The growth ofeach cluster was stopped when it reaches the edges ofthe square computational domain, 2 ℓ max × ℓ max , with aprescribed scale ℓ max . As a result, the number of particlesin various clusters is not identical. We generated 100clusters with ℓ max = 16 that have the minimal and themaximal number of particles 41 003 402 and 51 514 999,respectively. We also generated one larger cluster with145 199 976 particles by setting ℓ max = 17 (Fig. 1). Toour knowledge, this is the largest on-lattice DLA clusterever generated (in contrast, off-lattice DLAs of similarsizes have been earlier reported, e.g., [38]).
Fractal dimension versus growth rate
Knowing the history of growth of each generated clus-ter, we compute two conventional characteristics: thecluster radius, R ( t ), and the radius of gyration, R ( t ),as functions of the cluster size t (i.e., the number of par-ticles) R ( t ) = max ≤ k ≤ t (cid:26)q x k + y k (cid:27) , (1) R ( t ) = t t X k =1 ( x k + y k ) ! , (2)where ( x k , y k ) are the coordinates of the k -th attachedparticle (with the seed point of the cluster, ( x , y ), be-ing located at the origin). We checked that R ( t ) and R ( t ) behave similarly, being different by a factor be-tween 2 and 3. For this reason, we focus on the radiusof gyration which exhibits less fluctuations. Figure 2aillustrates a power law growth of R ( t ) with the clustersize t for one DLA cluster. For this cluster, we get thegrowth exponent β = 0 . −1 cluster size r ad i u s (a) scale N bo x e s (b) FIG. 2: (Color online) (a)
Radius of gyration R ( t ) as afunction of the cluster size t for one cluster (symbols). Alinear fit at loglog scale (line), ln R ( t ) = 0 .
594 ln t − . to 2 . (b) Number of non-empty boxes, N ℓ , at scale ℓ , for the same cluster. Solid lineshows a linear fit at loglog scale, ln N ℓ = − .
664 ln ℓ + 18 . and 2 . DLA, 0 .
583 [39]. For the same cluster, we also computeits box-counting dimension by evaluating the number N ℓ of non-empty boxes at scale ℓ , ranging from 1 (the sizeof one particle) to 2 ℓ max (the size of the whole cluster).Figure 2b shows a power law scaling N ℓ ∝ ℓ − D , whichenables us to determine the Minkowski (box-counting)dimension D . For this DLA cluster, we get D ≈ . /β ≈ .
684 by 1%. In order to check the relevance ofthis difference, we repeated the above analysis for 100 in-dependently generated clusters. We obtain the empiricalmean and standard deviation for two exponents: D = 1 . ± . , β = 0 . ± . . (3)The average fractal dimension D is smaller than the aver-age of the inverse of the growth rate, 1 /β = 1 . ± . . D = 1 /β for on-lattice DLA clusters. We also foundthat the box-counting fractal dimension D is remarkablyclose to the theoretical value 5 / Role of anisotropy
Various measures have been introduced in 1980’s tocharacterize the anisotropy of DLA clusters on the squarelattice [17, 25–28]. We propose another quantity, the angular growth rate , which is much better adapted to thestudy of anisotropic but mostly star-like structures. Wecover the plane by n s equal sectors S , . . . , S n s (of angle2 π/n s ) centered at the origin (the center of the cluster),and define the angular radius of gyration up to clustersize t : R θ ( t ) = n θ ( t ) t X k =1 ( x k + y k ) I ( x k ,y k ) ∈ S θ ! , (4)where I ( x k ,y k ) ∈ S θ is equal 1 if the point ( x k , y k ) belongsto the sector S θ of a discretized polar angle θ , and zerootherwise, while n θ ( t ) is the number of cluster pointsbelonging the sector S θ up to t . The angular growthrate, β θ , is defined from the expected power law scal-ing: R θ ( t ) ∝ t β θ as t → ∞ . In this way, one can probewhether the growth rate depends on the direction and,in particular, whether the growth rates along the squarelattice axes and along the diagonals are different.The left column of Fig. 3 shows the progressive growthof the largest DLA cluster shown in Fig. 1. One canclearly see how an isotropic structure of a small clus-ter (with 10 particles) slowly evolves into the cross-like anisotropic structure of larger clusters (e.g., with10 particles). For comparison, the right column ofFig. 3 shows a grayscale representation of the densityof points, averaged over 100 DLA clusters, at the same t . The average density is defined as the sum of indicatorfunctions of 100 independently generated clusters. Forsmall clusters ( t = 10 and below), the density is almostisotropic, meaning that the typical cluster has almost around shape. For larger clusters with t = 10 , the di-amond shape emerges, indicating a directional preferen-tial growth along the four axes. At t = 10 and t = 10 ,the diamond shape progressively transforms into a cross-like shape. These features are particularly well seen bylooking at two contours: the outer contour showing themaximally distant points from the center, and the innercontour showing the angular radius of gyration. Thesetwo contours were computed by identifying the points ofall 100 clusters that lie within a sector between angles θ and θ + δθ (with the angular resolution δθ = 1 ◦ , i.e., n s = 360). In each sector, the distance between the cen-ter and the most distant point, and the angular radius ofgyration R θ ( t ), were computed and then plotted versusthe polar angle θ from 0 to 360 ◦ . The outer and innercontours illustrate respectively the positions of extremeand average points of DLA clusters. Remarkably, thesetwo contours evolve with the cluster size in a very simi-lar way. Note that the evolution of a commonly observeddiamond-like structure of the square DLA clusters into a (a) t = 10 (b) t = 10 (c) t = 10 (d) t = 10 (e) t = 10 (f) t = 10 (g) t = 10 (h) t = 10 FIG. 3: (Color online).
Left column:
A DLA cluster atvarious cluster sizes t : 10 , 10 , 10 , and 10 , from the top tothe bottom (coarse-grained 512 ×
512 pictures);
Right col-umn:
Grayscale representation of the density of points, aver-aged over 100 DLA clusters, at the same t ; thin outer contourshows the maximally distant points from the center (an an-gular version of the maximal distance R defined by Eq. (1));thick inner contour shows the angular radius of gyration, withthe angular resolution of 1 ◦ . cross-like shape with four distinct arms was first conjec-tured by Meakin [27] and then confirmed by numericalsimulations on larger clusters in [24]. Moreover, the scal-ing exponents for the length and width of the four mainarms were claimed to be different [16, 19, 24]. We notehowever that the arguments elaborated in these papersrely on (over)simplified assumptions (e.g., the diamond-like limiting shape of DLA clusters), whereas predictionsof the scaling exponents were sometimes different. Whilethere was not doubt about anisotropic character of theon-lattice DLA growth, its explanations remained rathercontroversial.After this visual inspection, we proceed to quantify theanisotropic effects. Figure 4a shows how the angular ra-dius of gyration depends on the direction θ at differentcluster sizes t . One can see how the anisotropy is pro-gressively established (with four maxima along axes andfour minima along diagonal directions). For comparison,Fig. 4b shows the angular radius of gyration R θ ( t ) for anaverage cluster obtained by superimposing 100 clusters.As expected, this plot resembles that for one cluster butthe average over 100 clusters yields smoother curves. Theemergence of anisotropy is particularly clear at semilog-arithmic scale (Fig. 4c): flat profiles of R θ ( t ) versus θ atsmall cluster sizes t progressively become uneven, withprominent peaks in four axial directions.In order to reveal the different growth along axes anddiagonals, we aggregate the angular radii of gyration R θ ( t ) for 4 directions of square lattice axes to define R axis ( t ), and for 4 diagonal directions to define R diag ( t ).Since the number of points in each sector is significantlysmaller than in the whole cluster, fluctuations are muchstronger. For their reduction, we choose relatively largesectors of angle 11 . ◦ (with n s = 32) and we average theaggregated radii over 100 DLA clusters. The resultingaxial and diagonal radii R axis ( t ) and R diag ( t ) are shownin Fig. 5a. One can see the faster growth along the axesthan along the diagonals, with the growth rates 0 .
612 and0 . β θ obtained by linear fits at loglog scale of R θ ( t ) versus t (to reduce fluctuations, the angular radiusof gyration was averaged over 100 DLA clusters). Weobserve variations of β θ from 0 .
53 to 0 .
61, the minimaland maximal growth rates corresponding to the diagonalsand to the axes, respectively.
DISCUSSION
With the aid of extensive numerical simulations, wehave shown that DLA clusters on the square lattice ex-hibit strong anisotropic behavior driven by the local ag-gregation rules. In particular, the growth rate dependson direction, with the maximal growth rate along theaxes and the minimal one along the diagonals. This im-plies that after rescaling by cluster’s diameter, the mean
45 90 135 180 225 270 315 3600.511.52 x 10 angle r ad i u s (a) angle r ad i u s (b) angle r ad i u s (c) FIG. 4: (Color online) (a)
Angular radius of gyration R θ ( t )as a function of the angle θ for one cluster, at cluster sizes t = 2 , , . . . , (corresponding curves are arranged frombottom to top). (b,c) Angular radius of gyration R θ ( t ) as afunction of the angle θ , averaged over 100 clusters, at clustersizes t = 2 , , . . . , , at linear (b) and semilogarithmic (c) scales. For all plots, we set n s = 360. size of the cluster in all non axial directions converges tozero hence the scaling limit becomes deterministic andone-dimensional. On the other hand, on all scales fromthe particle size to the size of the entire cluster it has non-trivial box-counting fractal dimension which correspondsto the overall growth rate of the cluster. The latter is asort of a (non-arithmetic) average of the angular growthrates. This average fully ignores distinctions betweengrowth rates in different directions and thus partly ne-glects anisotropic effects, in particular, the fastest growthalong the axes. This suggests that the fractal nature of −1 cluster size r ad i u s (a) axesdiagonals angle β θ (b) FIG. 5: (Color online) (a)
Aggregated angular radiiof gyration for 4 axes, R axis ( t ), and for 4 diagonals, R diag ( t ), averaged over 100 DLA clusters, at cluster sizes t = 1 , , , . . . , . Linear fits at loglog scale (lines) are:ln R axis ( t ) = 0 .
612 ln t − .
923 and ln R diag ( t ) = 0 .
535 ln t − . (b) The angular growth rate β θ as a function of the an-gle θ , obtained from linear fits at loglog scale of R θ ( t ) versus t . For both plots, we set n s = 32. the lattice DLA should be understood in terms of fluctu-ations around one-dimensional backbone of the cluster.The crucial impact of the lattice anisotropy on theDLA growth naturally disappears for the intrinsicallyisotropic off-lattice DLA that had also been extensivelyinvestigated (see [38–44] and references therein). Thisrises an important question: how to explain the qual-itative difference between the on-lattice and off-latticeDLA? There are two obvious distinctions. First, thegrowth of the on-lattice DLA is controlled by the dis-crete harmonic measure versus the continuous one forthe off-lattice model. We believe that this is not reallyan issue, since the discrete measure converges rapidly tothe continuous one as the size of the aggregate increases[45, 46]. The second reason is that the models have dif-ferent rules for the local particle attachment. The effectof the local rules on the anisotropy of the clusters hasbeen observed before [47]. Given a cluster, the distribu-tion of places where a new particle will get close to thecluster is almost the same for particles performing ran-dom walk and for particles performing Brownian motion.By the concentration of the harmonic measure, the par-ticle which is started near the cluster will attach to thecluster very close to its starting position, with a largeprobability. The main difference is that for the latticethere are very few places where the particle can attach,especially in the vicinity of a “tip” in the cluster. Thishas two consequences: (i) the on-lattice particle is morelikely to attach to the tip in the direction of the fastestgrowth than the Brownian particle would do, and (ii) agrowing tip is less likely to be split into two competingbranches. As a consequence, the branches of off-latticeDLA are more wiggly.We conclude that the naive but widely used scalinglimit of on-lattice DLA fails due to anisotropy. The fu-ture analysis needs to account for anisotropic effects andto potentially focus on individual branches of large DLAclusters. Our results present thus the first step towardsfinding a proper rescaling of DLA clusters that is cru-cial to understand the fractal properties of the on-latticeDLA model and its scaling limit. Acknowledgments
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