6th_Sem_02 Plotting variation of E & Cv with T for BE & FD Statistics for 2 particles in 2 single state system using Python & Matplotlib

Plotting variation of Energy(E) & Specific heat at constant Volume(C_v) with Temperature(T) for Maxwell-Boltzmann & Fermi-Dirac Statics for 2 particles in 2 single state system using Python & Matplotlib 

FOR MB STATISTICS

Since the particles are identical in MB statistics, i.e. bosons are indistinguishable,

Configuration		Energy of the configuration (Ei)	Number of states (gi)
0	𝝴
AA	-	0	1
-	AA	2𝝴	1
A	A	𝝴	1

The Partition function of the system is,

The mean energy of the system,

Now,

Now,

or,

for simplification let us assume,                      𝟄=1 & 

So now,     

Now it's time to write a program that can plot these functions, you can use this program or can modify it for more customisation.

import numpy as np import matplotlib.pyplot as plt x=np.linspace(0.01,2,1000) z=1+np.exp(-2/x)+np.exp(-1/x) E=(2*np.exp(-2/x)+np.exp(-1/x))/z Ep=E/2.0 Esq=E**2 E2=(4*np.exp(-2/x)+np.exp(-1/x))/z Cv1=(E2-Esq)/x**2 Cv1p=Cv1/2.0 Cv2=np.diff(E)/np.diff(x) Cv2p=Cv2/2.0 xx=0.5*(x[1:]+x[0:-1]) plt.plot(x,Ep,'b',x,Cv1p,'r',xx,Cv2p,'g--') plt.xlabel('Temperature(T)') plt.ylabel('E & Cv') plt.title('E & Cv Vs. Temp. plots') plt.grid() plt.legend(['E','Cv1','Cv2']) plt.show()


Output:

This is the output of the program

FOR FD STATISTICS

The fermions are indistinguishable & they obey Pauli exclusion principle, so no two fermions can have the same quantum state.

Configuration		Energy of the configuration (Ei)	Number of states (gi)
0	є
A	A	є	1

So the simplified Partition function & energy functions are,

The program for plotting the curve is,

import numpy as np import matplotlib.pyplot as plt x=np.linspace(0.01,2,1000) z=np.exp(-1/x) E=(np.exp(-1/x))/z Ep=E/2.0 Esq=E**2 E2=(np.exp(-1/x))/z Cv1=(E2-Esq)/x**2 Cv1p=Cv1/2.0 Cv2=np.diff(E)/np.diff(x) Cv2p=Cv2/2.0 xx=0.5*(x[1:]+x[0:-1]) plt.plot(x,z,'k',x,Ep,'b',x,Cv1p,'r',xx,Cv2p,'g--') plt.xlabel('Temperature(T)') plt.ylabel('z, E & Cv') plt.title('z, E & Cv Vs. Temp. plots') plt.grid() plt.legend(['z','E','Cv1','Cv2']) plt.show()

Output:

This is the output of the program

If you change the program, post your program in the comments. Believe me it is very interesting to modify and create your own program.

Thanks for reading. If you have any questions, comment below. I'll try to help it.

🙏Thank You🙏