Help for simulation class/ Probability Distribution

[springsilver]

Hello,
I am new to python and I am taking a computational simulation class and my professor suggested to me to use python. I am an ESL, maybe I do not understand what does he want from me for the homework. Here is the question:
For each thermostat, do the following:
a) Plot the trajectory of the particle movement in x-y plane.
b) Test your two thermostat implementations by making a histogram of the probability of nding the
system in a small window around x=0.5 as a function of y. Compare your histograms to canonical
distribution (plot canonical distribution along with the histogram).
Implement the following thermostats:
I am able to draw the plot for the part a. However, I could not draw the right histogram for part b. The professor showed in the class that the probability distribution should also include a Boltzmann distribution line. I attached my codes. I hope that somebody helps me to clarify this situation. Thank you in advance.

import numpy as np
import matplotlib.pyplot as plt

t = 0                                   #initial time
N = 10                              #duration of simulation
dt = 0.01                               #timestep of integration
m = 1                                   #mass of a single single particle

x1 = 0.5
vx1 = 0.3
y1 = 0
vy1 = 0.3
                     
while t <= N:
    Fx0=(2*(np.pi)*np.sin(2*(np.pi)*x1))*((4*y1) + 1)
    Fy0=(-4*np.cos(2*(np.pi)*x1)) - (4*y1*((np.pi)**2))
    x1 = x1 + dt*vx1 + ((dt**2)*0.5*(Fx0/m))         #positons at current time
    y1 = y1 + dt*vy1 + ((dt**2)*0.5*(Fy0/m))
    Fx=2*(np.pi)*np.sin(2*(np.pi)*x1)*((4*y1) + 1)
    Fy=(-4*np.cos(2*(np.pi)*x1)) - (4*y1*((np.pi)**2))
    vx1 = vx1 + 0.5*dt*(Fx+Fx0)/m                   #calculate new velocities
    vy1 = vy1 + 0.5*dt*(Fy+Fy0)/m
    
    plt.scatter(x1,y1)                                # Plot function
    t = t + dt        

plt.title('Particle Movement')
plt.xlabel('x')
plt.ylabel('y')
plt.savefig('hw3_traj.tiff',dpi=300)
plt.show()