Metropolis X2X4.py

From Werner KRAUTH

Context

This page is part of my 2024 Beg Rohu Lectures on "The second Markov chain revolution" at the Summer School "Concepts and Methods of Statistical Physics" (3 - 15 June 2024).

Python program

import math
import random
import matplotlib.pyplot as plt
def u(x):
    return x ** 2 / 2.0 + x ** 4 / 4.0

x = 0.0
delta = 0.1

data = []
n_samples = 10 ** 6
for i in range(n_samples):
    new_x = x + random.uniform(-delta, delta)
    delta_u = u(new_x) - u(x)
    if random.random() < math.exp(-delta_u): x = new_x
    data.append(x)

plt.title('Metropolis algorithm, anharmonic oscillator' )
plt.xlabel('$x$')
plt.ylabel('$\pi(x)$')
plt.hist(data, bins=100, density=True,label='data')
XValues = []
YValues = []
for i in range(-1000,1000):
    x = i / 400.0
    XValues.append(x)
    YValues.append(math.exp(- u(x)) / 1.93525)
plt.plot(XValues, YValues, label='theory')
plt.legend(loc='upper right')
plt.show()

Further information

References

• Tartero, G., Krauth, W. Concepts in Monte Carlo sampling, Am. J. Phys. 92, 65–77 (2024) arXiv:2309.03136