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Kernel: Python 3 (Anaconda 2022)

The Random Walk

The first cell loads two external modules

import random
import matplotlib.pyplot as plt

Now we create a single random walk

x=[0]
for j in range(10):
    n=random.randint(0,1)
    if n==1:
        x.append(x[j]+1)
    else:
        x.append(x[j]-1)

plt.plot(x)
plt.xlabel('n')
plt.ylabel('displacement')
plt.show()
Image in a Jupyter notebook

More walks on one page

for run in range(500):
    x=[0]
    for j in range(1000):
        n=random.randint(0,1)
        if n==1:
            x.append(x[j]+1)
        else:
            x.append(x[j]-1)
    plt.plot(x)
plt.show()
Image in a Jupyter notebook

To investigate the average distance & displacement, we create a number of subloops. We end with two lists of the averages of many loops of different lengths

displacements=[]
distances=[]
for m in range(100):
    rundistances = []
    rundisplacements = []
    for run in range(1000): # do 1000 runs, each m steps long
        d=0
        for j in range(m):
            n=random.randint(0,1)
            if n==1:
                d+=1
            else:
                d-=1
        rundisplacements.append(d)
        rundistances.append(abs(d))
    displacements.append(sum(rundisplacements)/len(rundisplacements))
    distances.append(sum(rundistances)/len(rundistances))
plt.plot(distances,'rx',label='distance')
plt.plot(displacements,'bx',label='displacement')
plt.legend()
plt.show()
Image in a Jupyter notebook

The average displacement remains at zero, as expected. However, the average distance from the origin increases. This appears to be a square-root relationship.

import numpy as np
distancesSqd = [i**2 for i in distances]
x=np.arange(0,100)
y=distancesSqd
fit = np.polyfit(x,y,1)
fit_fn = np.poly1d(fit)
plt.plot(x,y, 'rx', x, fit_fn(x), '--k')
plt.xlabel('n')
plt.ylabel('distance squared')
plt.show()
Image in a Jupyter notebook