Physics 756 Assignment 2
1. Growth models
Choose one of the following growth models that we discussed in the lectures: Eden model, Ballistic deposition, RSOS model, or Random deposition with surface diffusion.
For your chosen model, carry out a simulation with a surface of length L with periodic boundary conditions, beginning with a flat surface. Measure the width of the interface as a function of L and t (you will have to average over repeat runs for each value of L). Show that it follows a dynamic scaling law:
You can do this by plotting as a function of x, where . Curves from different values of L should superimpose if you choose the right values of and z.
2. Diffusion model
Write a program to simulate the model for diffusion in 1 dimension. Sites on a line are numbered from 0 to L. The transition probabilities in one time step are:
P(n,t) is the probability of being at site n at time t. Use L = 200, = 0.05, and t = 1 unit. Begin with P(30,0) = 1, and P(n,0) = 0 for n 30. Use absorbing boundaries at 0 and L. The following recursion can be used to calculate P(n,t) at subsequent times:
Calculate P(n,t) after 4000 and after 100000 time steps and plot graphs of these distributions.
The solution to the diffusion equation can be written as a sum of sine wave functions (as in the lecture). Plot this solution for the two times above on the same graph as the probability distribution obtained from the simulations. Show that the diffusion equation in continuous time and space is a very good approximation of the model with discrete sites and discrete time steps.
3. Wright- Fisher Model
Consider the Wright Fisher model with 2N = 200, with a mutation rate u between two alleles that is equal in both directions. Assume that there is no selection. The current frequency of the X allele is x = m/2N. The probability that any one gene copy in the next generation is an X is
The probability that there are n copies of X in the next generation is
Begin with P(30,0) = 1, and P(n,0) = 0 for n 30.
The probability distribution at subsequent generations is given by
Choose two different values of u - one for which the stationary distribution is expected to be bell-shaped and one for which it is U-shaped. For each of these values of u, plot the probability distribution at several different times and show that it does converge to a stationary distribution at large times.
From the Kolmogorov forward equation, the stationary solution should be
where = 4Nu. Plot this function on the same graph as the stationary distribution from the simulations and show that it gives a very good approximation to the simulation.