Chapter 13 - Davidson Physics

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CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 556// s e t s t h e l a s t p a r t i c l e o f l a r g e r c l u s t e r to t h e l a s t p a r t i c l e o f s m a l l e r c l u s t e rl a s t P a r t i c l e [ l a r g e r C l u s t e r L a b e l ] = l a s t P a r t i c l e [ s m a l l e r C l u s t e r L a b e l ] ;// adds mass o f s m a l l e r c l u s t e r to t h e l a r g e r c l u s t e rmass [ l a r g e r C l u s t e r L a b e l ] += mass [ s m a l l e r C l u s t e r L a b e l ] ;To complete the merger, all the entries in site[x][y] corresponding to the smaller cluster arerelabeled with the label for the larger cluster, and the last cluster in the list is relabeled by thelabel of the small cluster, so that if there are n clusters they are labeled by 0, 1, . . . n − 1.a. Write a target class for class CCA. The class assumes that the diffusion coefficient is independentof the cluster mass. Choose L = 50 and N = 500 and describe the qualitative appearance ofthe clusters as they form. Do they appear to be fractals? Compare their appearance to DLAclusters.b. Compute the fractal dimension of the final cluster. Use the center of mass, r cm , as the originof the cluster, where r cm = (1/N) ( ∑ i x i, ∑ i y i)and (xi , y i ) is the position of the ith particle.Average your results over at least ten final clusters. Do the same for other values of L andN. Are the clusters formed by cluster-cluster aggregation more or less space filling than DLAclusters?c. Assume that the diffusion coefficient of a cluster of s particles varies as D s ∝ s −1/2 in twodimensions. Let D max be the diffusion coefficient of the largest cluster. Choose a randomnumber r between 0 and 1 and move the cluster if r < D s /D max . Repeat the simulations inpart (a) and discuss any changes in your results. What effect does the dependence of D on shave on the motion of the clusters?References and Suggestions for Further ReadingWe have considered only a few of the models that lead to self-similar patterns. Use your imaginationto design your own model of real-world growth processes. We encourage you to read the researchliterature and the many books on fractals.R. C. Ball and R. M. Brady, “Large scale lattice effect in diffusion-limited aggregation,” J. Phys. A18, L809–L813 (1985). The authors discuss the optimization algorithm used in Project 14.17.Albert-László Barabási and H. Eugene Stanley, Fractal Concepts in Surface Growth, CambridgeUniversity Press (1995).J. B. Bassingthwaighte, L. S. Liebovitch, and B. J. West, Fractal Physiology Oxford UniversityPress (1994).D. Ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems,Cambridge University Press (2005).K. S. Birdi, Fractals in Chemistry, Geochemistry, and Biophysics, Plenum Press (1993).Armin Bunde and Shlomo Havlin, editors, Fractals and Disordered Systems, revised edition,Springer-Verlag (1996).
CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 557Fereydoon Family and David P. Landau, editors, Kinetics of Aggregation and Gelation, North-Holland (1984). A collection of research papers that give a wealth of information, pictures,and references on a variety of growth models.Fereydoon Family, Daniel E. Platt, and Tamás Vicsek, “Deterministic growth model of patternformation in dendritic solidification,” J. Phys. A 20, L1177–L1183 (1987). The authors discussthe nature of Laplace fractal carpets.Fereydoon Family and Tamás Vicsek, editors, Dynamics of Fractal Surfaces, World Scientific(1991). A collection of reprints.Fereydoon Family, Y. C. Zhang, and Tamás Vicsek, “Invasion percolation in an external field:Dielectric breakdown in random media,” J. Phys. A. 19, L733–L737 (1986).Jens Feder, Fractals, Plenum Press (1988). This text discusses the applications as well as themathematics of fractals.Gary William Flake, The Computational Beauty of Nature, MIT Press (2000).J.-M. Garcia-Ruiz, E. Louis, P. Meakin, and L. M. Sander, editors, Growth Patterns in PhysicalSciences and Biology, NATO ASI Series B304, Plenum (1993).Peter Grassberger, “Critical percolation in high dimensions,” Phys. Rev. E 67, 036101-1–4 (2003).The author uses the Leath algorithm to estimate the value of p c .Thomas C. Halsey, “Diffusion limited aggregation: A model for pattern formation,” Physics Today53 (11), 36 (2000).J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, Methuen (1964). The chapteron percolation processes discusses a growth algorithm for percolation.H. J. Herrmann, “Geometrical cluster growth models and kinetic gelation,” Physics Reports 136,153–224 (1986).Robert C. Hilborn, Chaos and Nonlinear Dynamics, second edition, Oxford University Press(2000).Ofer Malcai, Daniel A. Lidar, Ofer Biham, and David Avnir, “Scaling range and cutoffs in empiricalfractals,” Phys. Rev. E, 56, 2817–2828 (1997). The authors show that experimentalreports of fractal behavior are typically based on a scaling range that spans only 0.5–2 decadesand discuss the possible implications of this limited scaling range.Benoit B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman (1983). An influentialand beautifully illustrated book on fractals.Imtiaz Majid, Daniel Ben-Avraham, Shlomo Havlin, and H. Eugene Stanley, “Exact-enumerationapproach to random walks on percolation clusters in two dimensions,” Phys. Rev. B 30, 1626(1984).
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