Monte Carlo technique
[IMP.: Along with the plots I also want to see the programs you used]
1. Compute the value of pi using Monte Carlo technique.
Consider the square region [0,2]x[0,2]. Now generate pairs of random numbers corresponding to x & y coordinates of randomly chosen points within the square region (the rand function of MATLAB generates uniformly distributed random numbers in the interval [0,1] - so multiply by 2, to get random numbers in the interval [0,2]). Check to see if they fall in the circle centered at (1,1) and with a radius of 1. If, out of N generated random points, M points fall within the circle, the ratio M/N is the ratio of the area of the circle to the area of the square. Hence compute the value of pi.
For each N, compute pi at least 10 times and find the standard deviation and mean (use MATLABs mean and std functions).
Plot how the mean and standard deviation of the computed value of pi changes as N is increased (use N=100, 250, 500 ,750, 1000, 2000, 5000, 10000, 50000 and 100000). (You can use MATLABs errorbar function to plot the mean and standard deviation of the data points)
2. Determination of random packing fraction of circles.
Bernal was probably the first one to try to obtain the random packing fraction of spheres by using monte carlo simulation. In his Bakerian lecture (1962), Bernal says "I was able to produce ... a model with spheres derived from the simple process of choosing the coordinates of a succession of points strictly at random and keeping all of them that were not less than a certain distance from points already chosen." (p. 305)
Implement this method in two dimensions. Choose a square (having sides of length 10 units, say). Now try packing circles with radius of 0.5 units. Choose random points in the square and check whether they are at least 2 radii away from the centre of a previously chosen circle or at least 1 radii away from the boundaries. If they are then accept this as the coordinate of a new circle. Proceed in this manner until you find that after many iterations you cannot find the coordinate of a new circle. Plot how many iterations are needed before each new circle is accepted. Choose a judicious cut-off value of the iteration number based on this data. Compute the packing fraction.
Keeping the radius of the circle constant, increase the size of the square box (from 10 units go upto 1000 units in 5 steps). At each step compute the packing fraction. Try to also obtain the standard deviation for the smaller sized squares. Plot the mean and standard deviations of the packing fraction.
Note that Bernal says "This method became more amd more tedious and wasteful towards the end of the process and often as many as a thousand points had to be chosen for one to be accepted". So be patient ! :-)