phys 114 week 6 problem 9

Problem 9. constructing MC simulation of an Einstein solid in Python

import numpy as np import matplotlib.pyplot as plt beta = 1 # defines the temperature N = 1000 # number of particles e = np.zeros(N) # size N array, Nth element is the energy of Nth particle avg_e = [] # array to track the average energy throughout the simulation for i in range(100000): p = np.random.randint(0,N) # select random particle of interest move = np.random.randint(0,2) # trial move, coin toss: if 1, consider adding 1 energy. if 0, consider adding -1 energy prob = np.exp(-beta) # boltzmann factor, probability of successfully raising the energy if move == 1: r = np.random.random() # random number, r, to determine whether the move to add energy is successful if r > prob: e[p] = e[p] + 1 # accept the move if the probability is above e^(-b*delta*E) else: if e[p] > 0: e[p] -= 1 # accept the move if it lowers the particle's energy, but not below zero avg_e.append(np.mean(e)) # update the average energy array # plot histogram of energies plt.subplot(2,2,1) plt.hist(e,bins=50) plt.title('histogram of energies') plt.xlabel('energy') # plot average energy throughout the simulation plt.subplot(2,2,2) plt.plot(range(len(avg_e)), avg_e) plt.title('average energy over iterations') plt.xlabel('iterations') plt.ylabel('average energy') #plt.figure(figsize=(10,10)) #plt.figure(figsize=(20,15)) plt.tight_layout() plt.show()

The two quantities I tracked throughout this MC simulation were the probability distribution of energies (histogram of energies) and the average energy throughout the simulation (average energy over iterations). Based on the first plot, we can see a very clear decaying exponential relationship for the energy values. This confirms the dependence of the probability distribution of energies on the Boltzmann factor. And for the average energy, we can see that it increases until aroun 1.5. Since beta is set to 1, this matches the 3/2 * kT prediction of the equipartition theorem.

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