Chapter 13 - Davidson Physics
Chapter 13 - Davidson Physics Chapter 13 - Davidson Physics
CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 552The sum in (14.20) is over all the boxes and involves the probabilities raised to the qth power. Forq = 0, we haveln N(l)D 0 = − lim . (14.21)l→0 ln lIf we compare the form of (14.21) with (14.17), we can identify D 0 with the box dimension. Forq = 1, we need to take the limit of (14.20) as q → 1. Letu(q) = ln ∑ ip i q , (14.22)and do a Taylor-series expansion of u(q) about q = 1. We haveu(q) = u(1) + (q − 1) dudq + . . . (14.23)The quantity u(1) = 0 because ∑ i p i = 1. The first derivative of u(q) is given by∑dudq = i p i q ln p∑ ii p i q = ∑ p i ln p i , (14.24)iwhere the last equality follows by setting q = 1. If we use the above relations, we find that D 1 isgiven by∑iD 1 = limp i ln p i. (information dimension) (14.25)l→0 ln lD 1 is called the information dimension because of the similarity of the p ln p term in the numeratorof (14.24) to the information form of the entropy.It is possible to show that D 2 as defined by (14.20) is the same as the mass dimension defined in(14.1) and the correlation dimension D c . That is, box counting gives D 0 and correlation functionsgive D 2 (cf. Sander et al.).There are many objects in nature that differ in appearance but have similar fractal dimension.An example is the different visual appearance in three dimensions of diffusion-limited aggregationclusters and the percolation clusters at the percolation threshold. (Both objects have a fractaldimension of approximately 2.5.) In some cases this difference can be accounted for by the multifractalproperties of an object. For multifractals the various D q are different, in contrast tomonofractals for which the different measures are the same. Percolation clusters are an exampleof a monofractal, because p i ∼ l D0 , the number of boxes N(l) ∼ l −D0 , and from (14.20), D q = D 0for all q. Multifractals occur when the growth quantities are not the same throughout the object,as frequently happens for the strange attractors produced by chaotic dynamics. Diffusion-limitedaggregation is an example of a multifractal.14.6 ProjectsAlthough the kinetic growth models we have considered yield beautiful pictures, there is much wedo not understand. For example, the fractal dimension of DLA clusters can be calculated only byapproximate theories whose accuracy is unknown. Why do the fractal dimensions have the values
CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 553that we estimated by various simulations? Can we trust our numerical estimates of the variousexponents, or is it necessary to consider much larger systems to obtain their true asymptotic values?Can we find unifying features for the many kinetic growth models that presently exist? What isthe relation of the various kinetic growth models to physical systems? What are the essentialquantities needed to characterize the geometry of an object?One of the reasons that kinetic growth models are difficult to understand is that the finalcluster typically depends on the history of the growth. We say that these models are examplesof “nonequilibrium behavior.” The combination of simplicity, beauty, complexity, and relevanceto many experimental systems suggests that the study of fractal objects will continue to involve awide range of workers in many disciplines.Project 14.15. The percolation cluster size distributionUse the Leath algorithm to determine the critical exponent τ of the cluster size distribution, n s ,for percolation clusters at p = p c :n s ∼ s −τ . (s ≫ 1) (14.26)Modify class SingleCluster so that many clusters are generated and n s is computed for a givenprobability p. Remember that the number of clusters of size s that are grown from a seed is theproduct sn s , rather than n s itself (see Problem 14.3a). Grow at least 100 clusters on a squarelattice with L ≥ 61. If time permits, use bigger lattices and average over more clusters, and alsoestimate the accuracy of your estimate of τ. See Grassberger for a discussion of an extension ofthis approach to estimating the value of p c in higher dimensions.Project 14.16. Continuum DLAa. In the continuum (off-lattice) version of diffusion-limited aggregation. the diffusing particlesare assumed to be disks of radius a. A disk executes a random walk until its center is withina distance 2a of the center of a disk that is already part of the DLA cluster. At each stepthe walker changes its position by (r cos θ, r sin θ), where r is the step size, and θ is a randomvariable between 0 and 2π. Modify your DLA program or class DLAApp to simulate continuumDLA.b. Compare the appearance of a continuum DLA cluster with a DLA cluster generated on asquare lattice. It is necessary to grow very large clusters (approximately 10 6 particles) to seethe differences.c. Use the mass dimension to estimate the fractal dimension of continuum DLA clusters andcompare its value with the value you found for the square lattice.Project 14.17. More efficient simulation of DLATo improve the efficiency of the algorithm, the walker in class DLAApp is restarted if it wonders toofar from the existing cluster. When the walker is within the distance startRadius of the seed,no optimization is used. Because there can be many unoccupied sites within this distance, it isdesirable to use an additional optimization technique (see Ball and Brady). The idea is to choose
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CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 553that we estimated by various simulations? Can we trust our numerical estimates of the variousexponents, or is it necessary to consider much larger systems to obtain their true asymptotic values?Can we find unifying features for the many kinetic growth models that presently exist? What isthe relation of the various kinetic growth models to physical systems? What are the essentialquantities needed to characterize the geometry of an object?One of the reasons that kinetic growth models are difficult to understand is that the finalcluster typically depends on the history of the growth. We say that these models are examplesof “nonequilibrium behavior.” The combination of simplicity, beauty, complexity, and relevanceto many experimental systems suggests that the study of fractal objects will continue to involve awide range of workers in many disciplines.Project 14.15. The percolation cluster size distributionUse the Leath algorithm to determine the critical exponent τ of the cluster size distribution, n s ,for percolation clusters at p = p c :n s ∼ s −τ . (s ≫ 1) (14.26)Modify class SingleCluster so that many clusters are generated and n s is computed for a givenprobability p. Remember that the number of clusters of size s that are grown from a seed is theproduct sn s , rather than n s itself (see Problem 14.3a). Grow at least 100 clusters on a squarelattice with L ≥ 61. If time permits, use bigger lattices and average over more clusters, and alsoestimate the accuracy of your estimate of τ. See Grassberger for a discussion of an extension ofthis approach to estimating the value of p c in higher dimensions.Project 14.16. Continuum DLAa. In the continuum (off-lattice) version of diffusion-limited aggregation. the diffusing particlesare assumed to be disks of radius a. A disk executes a random walk until its center is withina distance 2a of the center of a disk that is already part of the DLA cluster. At each stepthe walker changes its position by (r cos θ, r sin θ), where r is the step size, and θ is a randomvariable between 0 and 2π. Modify your DLA program or class DLAApp to simulate continuumDLA.b. Compare the appearance of a continuum DLA cluster with a DLA cluster generated on asquare lattice. It is necessary to grow very large clusters (approximately 10 6 particles) to seethe differences.c. Use the mass dimension to estimate the fractal dimension of continuum DLA clusters andcompare its value with the value you found for the square lattice.Project 14.17. More efficient simulation of DLATo improve the efficiency of the algorithm, the walker in class DLAApp is restarted if it wonders toofar from the existing cluster. When the walker is within the distance startRadius of the seed,no optimization is used. Because there can be many unoccupied sites within this distance, it isdesirable to use an additional optimization technique (see Ball and Brady). The idea is to choose
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