Problem 6.4 - Part (d)

Write a program (including the random number generator) to plot the position as a function of time of a random walker in 1D that at each time step has an equal probability of making a step of ±1. Plot an ensemble of 10 trajectories, each 1000 points long, and overlay error bars of width 3σ(t) on the plot

import numpy as np
import matplotlib.pyplot as plt

# random number generator, using the example in (6.74)
def rng(Xn):
    return ((8121 * Xn) + 28411) % 134456

num_lines = 10
num_points = 1000
half_prob = 134456 / 2.0
Xn = 0.5

for i in range(num_lines):
    step = 0
    walk = []
    for i in range(num_points):
        Xn = rng(Xn)
        if Xn > half_prob:
            step +=1
        else:
            step -=1
        walk.append(step)
    plt.plot(walk)

# Error bars for all iteration
# plt.errorbar(range(num_points), np.zeros(num_points), yerr=(3*np.sqrt(range(num_points))))

# Error bars for every 100
t = range(0, num_points, 100)
err = 3*np.sqrt(range(0, num_points, 100))
plt.errorbar(range(0, num_points, 100), np.zeros(num_points / 100), yerr=3*np.sqrt(range(0, num_points, 100)))
plt.ylabel("Steps")
plt.xlabel("Number of Iterations")
plt.show()


    

Problem 6.3 - Part (a) Bit Sequence

Generate bit sequence for maximal 4 LSFR

# initial - Xn-1, Xn-2, Xn-3, Xn-4
init_past = [1,1,1,1]
past = init_past

seq_list = []
while True:
    Xn = (past[0] + past[3]) % 2
    seq_list.append([Xn] + past)
    past = [Xn] + past[0:3]    
    if past == init_past:
        break
        
print seq_list
        

[[0, 1, 1, 1, 1], [1, 0, 1, 1, 1], [0, 1, 0, 1, 1], [1, 0, 1, 0, 1], [1, 1, 0, 1, 0], [0, 1, 1, 0, 1], [0, 0, 1, 1, 0], [1, 0, 0, 1, 1], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 1], [1, 1, 0, 0, 0], [1, 1, 1, 0, 0], [1, 1, 1, 1, 0]]

Problem 6.2 (c)

# random number generator, using the example in (6.74)
# constrained to 8 bits
def rng(Xn):
    return ((8121 * Xn) + 28411) % 134456