Chapter 13 - Davidson Physics
Chapter 13 - Davidson Physics Chapter 13 - Davidson Physics
CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 532as follows. Suppose that an occupied site corresponds to an infected person. Initially there is asingle infected person and the four nearest neighbor sites (on the square lattice) correspond tosusceptible people. At the next time step, we visit the four susceptible sites and occupy (infect)each site with probability p. If a susceptible site is not occupied, we say that the site is immuneand we do not test it again. We then find the new susceptible sites and continue until either thedisease is controlled or reaches the boundary of the lattice. Convince yourself that this growthmodel of a disease generates a cluster of infected sites that is identical to a percolation cluster atprobability p. The only difference is that we have introduced a discrete time step into the model.Some of the properties of this model are explored in Problem 14.5.Problem 14.5. A simple epidemic modela. Explain why the simple epidemic model discussed in the text generates the same clusters as inthe percolation model. What is the minimum value of p necessary for an epidemic to occur?Recall that in one time step, all susceptible sites are visited simultaneously and infected withprobability p. Determine how n, the number of infected sites, depends on the time t (thenumber of time steps) for various values of p. A straightforward way to proceed is to modifyclass SingleCluster so that all susceptible sites are visited and occupied with probability pbefore new susceptible sites are found. In Chapter 15 we will learn that this model is an exampleof a cellular automaton.b. What are some ways that you could modify the model to make it more realistic? For example,the infected sites might recover after a certain time.Eden model. An even simpler example of a growth model was proposed by Eden in 1958to simulate the growth of tumors or a bacterial colony. Although we will find that the resultantmass distribution is not a fractal, the description of the Eden model illustrates the general natureof the fractal growth models we will discuss.Choose a seed site at the center of the lattice for simplicity. The unoccupied nearest neighborsof the occupied sites are the perimeter or growth sites. In the simplest version of the model, agrowth site is chosen at random and occupied. The newly occupied site is removed from the list ofgrowth sites and the new growth sites are added to the list. This process is repeated many timesuntil a large cluster of occupied sites is formed. The difference between this model and the simpleepidemic model is that all tested sites are occupied. In other words, no growth sites ever become“immune.” Some of the properties of Eden clusters are investigated in Problem 14.6.Problem 14.6. The Eden modela. Modify class SingleCluster so that clusters are generated on a square lattice according tothe Eden model. A straightforward procedure is to occupy perimeter sites with probabilityp = 1. The simulation should be stopped when the cluster just reaches the edge of the lattice.What would happen if we were to occupy perimeter sites indefinitely? Follow the procedure ofProblem 14.3 and determine the number of occupied sites M(r) within a distance r of the seedsite. Assume that M(r) ∼ r D for sufficiently large r, and estimate D from the slope of a log-logplot of M versus r. A typical log-log plot is shown in Figure 14.8 for L = 61. Can you concludefrom your data that Eden clusters are compact?
CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 533108ln M64200 1 2 3 4ln rFigure 14.8: Plot of ln M versus ln r for a single Eden cluster generated on a L = 61 square lattice.A least squares fit from r = 2 to r = 32 yields a slope of approximately 2.01.b. Modify your program so that only the perimeter or growth sites are shown. Where are themajority of the perimeter sites relative to the center of the cluster? Grow as big a cluster astime permits.Invasion percolation. A dynamical process known as invasion percolation has been used tomodel the shape of the oil-water interface that occurs when water is forced into a porous mediumcontaining oil. The goal is to use the water to recover as much oil as possible. In this processa water cluster grows into the oil through the path of least resistance. Consider a lattice of sizeL x × L y , with the water (the invader) initially occupying the left edge (see Figure 14.9). Theresistance to the invader is given by assigning to each lattice site a uniformly distributed randomnumbers between 0 and 1; these numbers are fixed throughout the invasion. Sites that are nearestneighbors of the invader sites are the perimeter sites. At each time step, the perimeter site withthe lowest random number is occupied by the invader and the oil (the defender) is displaced. Theinvading cluster grows until a path of occupied sites connects the left and right edges of the lattice.After this path forms, there is no need for the water to occupy any additional sites. To minimizeboundary effects, periodic boundary conditions are used for the top and bottom edges and allquantities are measured only over a central region for from the left and right edges of the lattice.Class Invasion implements the invasion percolation algorithm. The two-dimensional arrayelement site[i][j] initially stores a random number for the site at (i,j). If the site at (i,j)is occupied, then site[i][j] is set equal to 1. If the site at (i,j) is a perimeter site, thensite[i][j] is increased by 2. In this way we know which sites are perimeter sites and the valueof the random number is associated with the perimeter site. A new perimeter site is inserted intoits proper ordered position in the lists perimeterListX and perimeterListY. The perimeter listsare ordered so that the site with the largest random number is at the beginning.
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CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 533108ln M64200 1 2 3 4ln rFigure 14.8: Plot of ln M versus ln r for a single Eden cluster generated on a L = 61 square lattice.A least squares fit from r = 2 to r = 32 yields a slope of approximately 2.01.b. Modify your program so that only the perimeter or growth sites are shown. Where are themajority of the perimeter sites relative to the center of the cluster? Grow as big a cluster astime permits.Invasion percolation. A dynamical process known as invasion percolation has been used tomodel the shape of the oil-water interface that occurs when water is forced into a porous mediumcontaining oil. The goal is to use the water to recover as much oil as possible. In this processa water cluster grows into the oil through the path of least resistance. Consider a lattice of sizeL x × L y , with the water (the invader) initially occupying the left edge (see Figure 14.9). Theresistance to the invader is given by assigning to each lattice site a uniformly distributed randomnumbers between 0 and 1; these numbers are fixed throughout the invasion. Sites that are nearestneighbors of the invader sites are the perimeter sites. At each time step, the perimeter site withthe lowest random number is occupied by the invader and the oil (the defender) is displaced. Theinvading cluster grows until a path of occupied sites connects the left and right edges of the lattice.After this path forms, there is no need for the water to occupy any additional sites. To minimizeboundary effects, periodic boundary conditions are used for the top and bottom edges and allquantities are measured only over a central region for from the left and right edges of the lattice.Class Invasion implements the invasion percolation algorithm. The two-dimensional arrayelement site[i][j] initially stores a random number for the site at (i,j). If the site at (i,j)is occupied, then site[i][j] is set equal to 1. If the site at (i,j) is a perimeter site, thensite[i][j] is increased by 2. In this way we know which sites are perimeter sites and the valueof the random number is associated with the perimeter site. A new perimeter site is inserted intoits proper ordered position in the lists perimeterListX and perimeterListY. The perimeter listsare ordered so that the site with the largest random number is at the beginning.
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