Sample calculations from PHYS 211 entropy lab¶
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#from sympy.utilities.iterables import partitions
#from sympy import binomial
import numpy as np
from scipy.special import binom
import matplotlib as mpl
import matplotlib.pyplot as plt
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# Following is an Ipython magic command that puts figures in notebook.
%matplotlib notebook
# M.L. modifications of matplotlib defaults
# Changes can also be put in matplotlibrc file,
# or effected using mpl.rcParams[]
mpl.style.use('classic')
plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures
plt.rc('axes', labelsize=16, titlesize=14)
plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new size
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def omega(q,n):
'''Multiplicity of states of Einstein solid with n atoms and q units of energy'''
return binom(q+n-1,q)
Define system parameters¶
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n_A = 100
n_B = 400
n_total = n_A + n_B
q_total = 400
Calculate multiplicities and entropies¶
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q = np.linspace(0, q_total,q_total + 1)
omega_A = omega(q,n_A)
omega_B = omega(q,n_B)
s_A = np.log(omega_A)
s_B = np.log(omega_B)
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plt.figure()
x = np.linspace(0, q_total, q_total+1)
# xB = energy_total - xD
yB = np.flip(s_B)
yA = s_A
plt.xlim(0,q_total)
plt.ylim(0,750)
plt.scatter(x , yA, label='$S_A$', marker='o', color='black')
plt.scatter(x, yB, label='$S_B$', marker='d', color='blue')
plt.scatter(x, yA+yB, label='$S_A+ S_B}$', marker='s', color='r')
plt.xlabel('$q_A$')
plt.ylabel('entropy')
plt.legend(loc='upper right')
plt.grid()
plt.axhline(0);
Calculate temperatures¶
\begin{eqnarray*} \frac{1}{T_i} &=& \left.\frac{dS}{dq}\right\vert_i \\ &\simeq& \frac{S_{i+1} - S_{i-1}}{2\epsilon} \end{eqnarray*}Solving for $T_i$ gives $$ T_i \simeq \frac{2\epsilon}{S_{i+1} - S_{i-1}}. $$
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t_A = 2/(s_A[3:len(s_A)] - s_A[1:len(s_A)-2])
t_B = 2/(s_B[3:len(s_B)] - s_B[1:len(s_B)-2])
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plt.figure()
x = np.linspace(1,q_total-2,q_total-2)
plt.scatter(x, t_A, color='black',label='$T_A$')
plt.scatter(x, np.flip(t_B), color='blue', label='T_B')
plt.xlim(0,q_total)
plt.ylim(0,5)
plt.grid()
plt.axhline(0)
plt.xlabel('$q_A$')
plt.ylabel('Temperature')
plt.legend(loc='best');
Version information¶
version_information is from J.R. Johansson (jrjohansson at gmail.com); see Introduction to scientific computing with Python for more information and instructions for package installation.
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%load_ext version_information
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version_information numpy, scipy, matplotlib
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