Dla diffusion limited aggregation


Dla diffusion limited aggregation

We now consider a modification of the random walk. A random walker in the plane moves at each step with equal probability either upwards, downwards, to the left, or to the right. Imagine that we start the walker at some randomly chosen site on the boundary of a square. If he walks out of the square, we start over with a new walker at a new randomly chosen site. At the center of the square is a fixed «seed». If the walker ever reaches a site which borders on the seed, the walker sticks to the seed and remains there. We then start with a new walker randomly chosen again on the boundary of the square. He walks until he either walks out of the square, or else reaches a site which borders on the now larger seed in which case he sticks and remains there. In this way we grow an object in the center of the square. Such a process is known as «diffusion limited aggregation», and the object that grows is called a «DLA cluster.»

After many such walkers stick on, what will be the shape of the growing seed? A natural guess is that, since walkers enter equally likely from all directions, one will grow a filled in circle, or perhaps a filled in square. Instead, one finds the following object shown below (this object is the result of 900 walkers sticking to the seed — the boundary in this example happened to be rectangular, not square, which is why the object is stretched more in the horizontal direction).

Such an object is called a fractal. A fractal is an object that looks the same on all length scales. For example, the fractal above consists of big branches which shoot off smaller branches, which shoot off still smaller branches, and so on. A fractal is also an object which can be thought of as having a non-integer dimension.

For example, suppose we measure length in units of «a» (you may think of «a» as being one inch, or one centimeter, or any other unit of length). Then in a line of length L, the number of such units that fill up the line is N = L/a

If we have a square with side of length L, and we measure area in units of a 2 , then the number of such units that fill up the square is N = L 2 /a 2

If we have a cube with side L, and we measure volume in units of a 3 , then the number of such units that fill up the cube is N = L 3 /a 3

In each case we found that the number of units that fill up the object is proportional to L D where L is the characteristic length of the object, and D is the dimension. For the fractal above, the number of units N that make up the fractal (in the DLA cluster this is just the number of attached walkers) scales as N

1.6 is a non-integer number. We can guess that for the DLA fractal 1


Calculating the properties of a DLA cluster, such as its fractal dimension, or the rate of growth at various points on its various branches, is very much a topic of current research in theoretical physics. The DLA cluster is an example of how very complex behavior can arise from very simple and easily understood basic rules.

Диффузия ограниченной агрегации — Diffusion-limited aggregation

Ограниченная диффузия агрегация (DLA) представляет собой процесс , в котором частица , проходящая в блуждание вследствие броуновское движения кластера вместе с образованием агрегатов таких частиц. Эта теория, предложенная Т. Виттен младшего и LM Sander в 1981 году, применяется к агрегации в любой системе , где диффузия является основным средством транспорта в системе. DLA можно наблюдать во многих системах , таких как электроосаждение, Хил-Шоу течение , полезные ископаемые, и диэлектрический пробой .

Кластеры , образованные в процессах DLA называется броуновскими дерева . Эти кластеры являются примером фрактала . В 2D эти фракталы проявляют размер около 1,71 для свободных частиц, которые неограниченный решеткой, однако компьютерное моделирование DLA на решетке изменит фрактальной размерности немного для DLA в той же размерности вложения . Некоторые вариации наблюдаются в зависимости от геометрии роста, будь то из одной точки в радиальном направлении наружу или от плоскости или линии, например. Два примера агрегатов , сгенерированные с использованием микрокомпьютера, позволяя случайных пешеходов придерживаться совокупности (первоначально (I) по прямой линии , состоящей 1300 частиц и (II) одну частицу в центре), показаны справа.

Компьютерное моделирование DLA является одним из основных средств изучения этой модели. Существует несколько методов для достижения этой цели . Моделирование может быть сделано на решетке любой желаемой геометрии размерности вложения (это было сделано в до 8 размеров) или моделирование может быть сделано больше вдоль линий стандартного молекулярной динамики моделирования , где частица может свободно блуждание до тех пор , пока он получает в течение определенного критического диапазона , после чего он вытягивается на кластер. Критической важности является то , что число частиц , проходящих броуновское движение в системе сохраняется очень низкий , так что только диффузный характер системы присутствует.

содержание

Работа на основе ограниченной диффузии агрегации

Сложные и органические формы , которые могут быть получены с алгоритмами агрегации ограниченной диффузии были изучены художниками. Simutils, часть toxiclibs с открытым исходным кодом библиотеки для языка программирования Java , разработанный Карстен Шмидт, позволяет пользователям применять процесс DLA на предварительно определенное руководящих принципов или кривых в моделировании пространства и через различные другие параметры динамически направлять рост 3D форм.

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Diffusion-limited aggregation


In diffusion-limited aggregation (DLA), particles undergo random walks due to Brownian motion. They cluster together to form aggregates.

DLA can be used to model systems such as lichen growth, the generation of polymers out of solutions, carbon deposits on the walls of a cylinder of a Diesel engine, path of electric discharge, and urban settlement.

In this simulation, the initial aggregate is set to be the bottom row of an NxN lattice. Particles are launched from a random cell in the top row. A particle’s random walk is set to have the following probabilities: up: 0.15, down: 0.35, and left or right: 0.25. The particle continues until it sticks to a neighbouring cell or leaves the lattice.

DLAs appear to be related to the Fibonacci sequence in terms of the branching sequence of the aggregate (see the references for more information about this, I haven’t looked into it in much detail). Additionally, small clusters evolve toward a five-branch symmetry. Note that the ratio of successive Fibonacci numbers approaches the golden ratio, which appears in the geometry of the pentagon.

Dla diffusion limited aggregation

An implementation of the Diffusion Limited Aggregation algorithm on a 2D grid with a central point attractor. A detailed description of the algorithm is present here. We also suggest a technique to estimate the stickiness factor k for a given aggregate from a DLA output.

The basic version of the algorithm can be found in the notebook Diffusion+Limited+Aggregation.ipynb . However, this is terribly slow. The running time increases exponentially as the size of the grid increases (since search space increases). To quicken the simulations, we make an alteration to the original algorithm. Initialization of a new particle is done at any point on the minimum bounding circle of the aggregate, rather than the square boundary of the grid. The implementation of DLA with this optimization can be found in Diffusion+Limited+Aggregation+Optimized.ipynb .

A proof of correctness for the alteration to DLA (under some assumptions) can be found in the file Optimization Proof.pdf .

Output of our DLA implementation at various number of particles. Stickiness factor 1.0, 0.5, 0.1 respectively.


The only requirements are numpy, matplotlib and joblib. A requirements.txt file has been provided. Running the following command should suffice.

pip install -r requirements.txt

You can either run the notebooks, or import the DLA class from dlaClass.py for your experiments.

We try to estimate stickiness by trying to find a nice propoerty of the aggregate (in our case, the surface area), which is monotonic and has a high correlation with the stickiness factor. We then try to model this property as a function of the stickiness factor k .

A detailed analysis can be found in DLA stickiness analysis.ipynb . Some exapmles of k estimations on unseen aggregates can be found in Estimate k.ipynb .

We recorded the value of surface area of the aggregate for different values of n and k. The (n, k, surface area) tuples are available for the following simulations

  • We ran 15 simulations for n in range(100, 20001, 100) , for values of k in linspace(1e-3, 5e-2, 40) . All this data can be found in the directories logFiles and logFiles2 .
  • We ran 2 simulations for n in range(100, 80001, 100) , for values of k in linspace(1e-3, 5e-2, 40) . All this data can be found in the directories logFiles80K .


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A simple model of kinetic growth process is diffusion-limited aggregation (DLA), which consists of particles in Brownian motion that «stick» together in a square lattice. In the 1D case, particles are added in random positions with the same value of height, which increases at each step. For the 2D case, other subtleties are involved; a particle is released from a randomly selected initial location within a circle of radius to move randomly. If the particle moves to a location contiguous to an occupied site, it is added to the cluster. The particle continues to move until it is captured or moves a distance away from the original position in the circle. When the distance is varied, the cluster can be develops into a fractal pattern, as described in the paper by Witten and Sander.

Contributed by: Enrique Zeleny (March 2011)
Open content licensed under CC BY-NC-SA

Diffusion-limited aggregation

In diffusion-limited aggregation (DLA), particles undergo random walks due to Brownian motion. They cluster together to form aggregates.

DLA can be used to model systems such as lichen growth, the generation of polymers out of solutions, carbon deposits on the walls of a cylinder of a Diesel engine, path of electric discharge, and urban settlement.


In this simulation, the initial aggregate is set to be the bottom row of an NxN lattice. Particles are launched from a random cell in the top row. A particle’s random walk is set to have the following probabilities: up: 0.15, down: 0.35, and left or right: 0.25. The particle continues until it sticks to a neighbouring cell or leaves the lattice.

DLAs appear to be related to the Fibonacci sequence in terms of the branching sequence of the aggregate (see the references for more information about this, I haven’t looked into it in much detail). Additionally, small clusters evolve toward a five-branch symmetry. Note that the ratio of successive Fibonacci numbers approaches the golden ratio, which appears in the geometry of the pentagon.

Dla diffusion limited aggregation

We now consider a modification of the random walk. A random walker in the plane moves at each step with equal probability either upwards, downwards, to the left, or to the right. Imagine that we start the walker at some randomly chosen site on the boundary of a square. If he walks out of the square, we start over with a new walker at a new randomly chosen site. At the center of the square is a fixed «seed». If the walker ever reaches a site which borders on the seed, the walker sticks to the seed and remains there. We then start with a new walker randomly chosen again on the boundary of the square. He walks until he either walks out of the square, or else reaches a site which borders on the now larger seed in which case he sticks and remains there. In this way we grow an object in the center of the square. Such a process is known as «diffusion limited aggregation», and the object that grows is called a «DLA cluster.»

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After many such walkers stick on, what will be the shape of the growing seed? A natural guess is that, since walkers enter equally likely from all directions, one will grow a filled in circle, or perhaps a filled in square. Instead, one finds the following object shown below (this object is the result of 900 walkers sticking to the seed — the boundary in this example happened to be rectangular, not square, which is why the object is stretched more in the horizontal direction).

Such an object is called a fractal. A fractal is an object that looks the same on all length scales. For example, the fractal above consists of big branches which shoot off smaller branches, which shoot off still smaller branches, and so on. A fractal is also an object which can be thought of as having a non-integer dimension.

For example, suppose we measure length in units of «a» (you may think of «a» as being one inch, or one centimeter, or any other unit of length). Then in a line of length L, the number of such units that fill up the line is N = L/a

If we have a square with side of length L, and we measure area in units of a 2 , then the number of such units that fill up the square is N = L 2 /a 2

If we have a cube with side L, and we measure volume in units of a 3 , then the number of such units that fill up the cube is N = L 3 /a 3


In each case we found that the number of units that fill up the object is proportional to L D where L is the characteristic length of the object, and D is the dimension. For the fractal above, the number of units N that make up the fractal (in the DLA cluster this is just the number of attached walkers) scales as N

1.6 is a non-integer number. We can guess that for the DLA fractal 1

Calculating the properties of a DLA cluster, such as its fractal dimension, or the rate of growth at various points on its various branches, is very much a topic of current research in theoretical physics. The DLA cluster is an example of how very complex behavior can arise from very simple and easily understood basic rules.

Diffusion-limited aggregation

Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, [1] is applicable to aggregation in any system where diffusion is the primary means of transport in the system. DLA can be observed in many systems such as electrodeposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown.

The clusters formed in DLA processes are referred to as Brownian trees. These clusters are an example of a fractal. In 2-D these fractals exhibit a dimension of approximately 1.71 for free particles that are unrestricted by a lattice, however computer simulation of DLA on a lattice will change the fractal dimension slightly for a DLA in the same embedding dimension. Some variations are also observed depending on the geometry of the growth, whether it be from a single point radially outward or from a plane or line for example. Two examples of aggregates generated using a microcomputer by allowing random walkers to adhere to an aggregate (originally (i) a straight line consisting 1300 particles and (ii) one particle at center) are shown on the right.

Computer simulation of DLA is one of the primary means of studying this model. Several methods are available to accomplish this. Simulations can be done on a lattice of any desired geometry of embedding dimension, in fact this has been done in up to 8 dimensions, [2] or the simulation can be done more along the lines of a standard molecular dynamics simulation where a particle is allowed to freely random walk until it gets within a certain critical range at which time it is pulled onto the cluster. Of critical importance is that the number of particles undergoing Brownian motion in the system is kept very low so that only the diffusive nature of the system is present.

Dla diffusion limited aggregation


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Topological properties of diffusion limited aggregation and cluster


Topological properties of diffusion limited aggregation and cluster

J. Phys. A: Math. Gen. 17 (1984) L975-L981. Rinted in Great Britain

LE’ITER TO THE EDITOR

Topological properties of diffusion limited aggregation and cluster-cluster aggregation P Meakint, I MajidS, S HavlinO and H Eugene Stanley$ t Central Research and Development Department, Experimental Station, E I DuPont de Nemours and Company Inc, Wilmington, DE 19898, USA t Centre for Polymer Studies11 and Department of Physics, Boston University, Boston, MA Oz215, USA 5 Division of Computer Research and Technology, National Institutes of Health, Bethesda, MD 20205, USA Received 11 September 1984

Abstract. The detailed topological or ‘connectivity’ properties of the clusters formed in diffusion limited aggregation (DLA) and cluster-cluster aggregation (CCA) are cons >
Cons >
/I Supported in part by grants from ONR and NSF. 0305-44701841180975+ 07$02.25 @ 1984 The Institute of Physics

Letter to the Editor

the length scale over which it is measured increases. If M ( R ) is the cluster mass within a Pythagorean distance R of a cluster point, and p ( R ) = M ( R ) / R d is the density, then one writes

p ( R ) — Rdf-d. (1) The fractal dimension concept has permitted extensive comparisons between large-scale computer simulations (e.g. Meakin 1983a, b, c, e) and several mean-field type theories (e.g. Muthukumar 1983, Tokuyama and Kawasaki 1984, Hentschel 1984, Hentschel and Deutch 1984). More recently, it has become possible to actually measure df for naturally-occurring aggregates and to compare the experimental values with results from simulations and from theory (see e.g. Forrest and Witten 1979, Nittmann et a1 1984, Weitz and Oliveira 1984, Niemeyer et a1 1984, Schaefer et a1 1984, Schaefer and Keefer 1984, Bale and Schm >

(see Middlemiss et a1 1980, Pike and Stanley 1981, Hong and Stanley 1983a,b, Herrmann et a1 1984). Equivalently, one may write


R -8; (2b) with v’= l/dmin(Havlin and Nossal 1984, Vannismenus et a1 1984). Since the minimum path in the cluster should not be shorter than the Pythagorean distance nor longer than a completely random walk between the two points, we expect 1sdmins2

Letter to the Editor

Figure 3. Dependence of the mass M ( P ) included within a minimum distance t ‘ for 2D and 4

cluster-cluster aggregates. In method 1 the origin site is considered to be a distance of 1 and the nearest-neighbour sites are considered to be at a distance P of 2. In method 2 both the origin site and its nearest neighbours are included in the mass at distance 1 and the next-nearest-neighbour sites are included in the mass at distance 2. Part (a) shows the results obtained from the ZD CCA model using method 2 ( D (M) MO). Part ( b ) shows the results obtained from the 4D clusters using both methods.

1984b), then the spectral dimension (see Stanley (1984) and references therein) is given by d s = 2 d p / ( d p + 1).

If de = df, then for all values of df we have

This result was also presented by Alexander (1983). Very recently, Aharony and Stauffer (1984) (AS) have argued, with one simple assumption, that (6b) should hold for any aggregate below the lower critical dimension d,. AS choose d , to be the dimension below which all the growth sites (Leyvraz and Stanley 1983) cannot fit into a thin annulus, and find d ; = 2 — a result also obtained by Coniglio and Stanley (1984) using quite different methods (see also Sahimi 1984). Recent work has questioned the assumption underlying the AS argument (Stanley et a1 1984b, Havlin 1984a, Hong 1984), but it does appear that (6b) holds for DLA for dr above d , (as well as below) for the reasons given in deriving ( 6 b ) :that DLA aggregates are loopless and have the property that dmin= 1 ( d e = df). It should be mentioned that a direct test of ( 6 6 ) is provided by extensive simulations (Meakin and Stanley 1983) of the probability of a

Letter to the Editor

random walk on a DLA cluster returning to the origin after t steps, Po the mean and the mean-square displacement (1’) t d s / d f . number of sites covered (s) Averaging the estimates obtained by these three calculations, one obtains d, = 1.25 0.10 ( d = 2 ) and d , = 1.35*0.12 ( d = 3 ) . The d = 2 result agrees well with (66) (d,=5/4) while the d = 3 result is within the error bars ( d , = 10/7). In summary, we have obtained results for the behaviour of dminand dp for DLA and CCA. By doing calculations for d = 2 , 3 , 4 we have found two striking results: (i) dmin= 1 for DLA, apparently independent of d, prov > 2 as well as d f

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