import numpy as np import matplotlib.pyplot as plt temps = [5, 50, 500] ks = [j for j in range(101)] listzs = [[], [], []] for i, temp in enumerate(temps): for k in ks: listzs[i].append(np.exp(-(k ** 2) / temp)) print(f"Temp = {temp}, listzs[i][100] = {listzs[i][100]}, ratio = {listzs[i][100] / listzs[i][1]}") plt.plot(ks, listzs[i], 'o', label=f"{temp}K") plt.ylabel(r'exp(-k^2/T)') plt.xlabel('k') plt.legend() plt.show()

sums = [] for i, temp in enumerate(temps): sums.append(np.sum(listzs[i])) print(f"Temp = {temp}, sum = {sums[i]}")

analyticals = [] for i, temp in enumerate(temps): analyticals.append(np.sqrt(np.pi * temp) / 2) print(f"analytical solution = {analyticals[i]}") #The analytical solution is a good approximation to the numerical evaluation for all values of T.

for i in range(len(temps)): print(f"relative error is {(analyticals[i] - sums[i]) / sums[i]}")

def sum_vs_int(k, T): '''k 1d array denoting the index of points to plot (assumes consecutive integers) T - parameter for sum''' #expfour = exponent(k, T) expfour = np.exp(-k*k/T) plt.plot(k, expfour, 'o') plt.bar(k, expfour, width=1, align='edge', alpha=0.4) plt.plot([k[0], k[0]], [0,expfour[0]], '--', color='tab:blue') for i in range(len(k)): plt.plot([k[i]+1,k[i]+1], [0,expfour[i]], '--', color='tab:blue') plt.plot([k[i],k[i]+1], [expfour[i],expfour[i]], '--', color='tab:blue') x = np.linspace(k[0],k[-1]+1,100) y = np.exp(-x*x/T) plt.plot(x, y, color='black') title = 'T = {}'.format(T) plt.title(title) plt.show()

firstTen = np.array([i for i in range(10)]) for temp in temps: sum_vs_int(firstTen, temp)