Assignment 8

General feedback

Overall, it was very good. The main issues were:

  • Some people were still forgetting to label axes and/or include a title and legend
  • Not initialising the walker correctly and therefore plotting the 1 dimensional case 4 times
  • A few people forgot to plot the line for the average squared position in Task 1
  • One common problem was that some students plotted the square of the average position, instead of the average of the squared position. The assignment asked for:

$\langle x^2\rangle = \frac{1}{ N_{\rm walker} }\sum\limits_{j=0}^{N_{\rm walker}} x_j^2.$

and not

$\langle x\rangle^2 = \frac{1}{ N_{\rm walker} }\left(\sum\limits_{j=0}^{N_{\rm walker}} x_j\right)^2.$

  • Not plotting a series of walker positions to show diffusion but the path of a single walker and making the first two plots with the walker index in the x axis instead of the step.

Plot examples

nsteps = 1000
ndim = 1
nwalkers = 100
distances = numpy.ndarray((nwalkers, nsteps+1))
distances2 = numpy.ndarray((nwalkers, nsteps+1))
for i in range(nwalkers):
    w=walker(numpy.zeros(ndim),ndim=ndim)
    ys=w.doSteps(nsteps)
    distances[i] = ys[:,0]
    dist2=numpy.array([sum(yy*yy) for yy in ys])
    distances2[i] = dist2
    
plt.plot(range(nsteps+1), numpy.mean(distances,0),label="$<x>$")
plt.plot(range(nsteps+1), numpy.mean(distances2,0),label="$<x^2>$")
plt.legend()
plt.title("average position and squared position")
plt.xlabel("number of steps")
plt.ylabel("average over 100 walkers")


nsteps=100
for ndim in range(1,5):
    nwalkers=400
    distances=numpy.ndarray((nwalkers,nsteps+1),dtype=float)
    for i in range(nwalkers):
        w=walker(numpy.zeros(ndim),ndim=ndim)
        ys=w.doSteps(nsteps)
        dist2=numpy.array([sum(yy*yy) for yy in ys])
        distances[i]=dist2
    plt.plot(range(nsteps+1),numpy.mean(distances,0),label="ndim={0}".format(ndim))
plt.legend()



ndim=2
nwalkers=500

rand_pos = numpy.random.uniform(size=(nwalkers,2))

plt.figure(figsize=(18,6))
for i, nsteps in enumerate([10,100,500]):
    plt.subplot(131+i)
    plt.xlim((-3,4))
    plt.ylim((-3,4))
    finalPosx = []
    finalPosy = []
    for i in range(nwalkers):
        w=walker(rand_pos[i],ndim=ndim, step_size=0.05)
        numpy.random.seed(i)
        ys=w.doSteps(nsteps)
        yst=ys.T
        x, y = ys[-1]
        finalPosx.append(x)
        finalPosy.append(y)
    plt.title("after {} steps".format(nsteps))    
    plt.xlabel("$x$")    
    plt.ylabel("$y$")    
    plt.title("after {} steps".format(nsteps))    
    plt.scatter(finalPosx, finalPosy, alpha=0.1)