ZigZag 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
time_ev = 0.0
sigma = 1 if x <= 0.0 else -1

data = []
n_samples = 10 ** 6
while len(data) < n_samples:
    delta_u = -math.log(random.random())
    new_x = sigma * math.sqrt(-1.0 + math.sqrt(1.0 + 4.0 * delta_u))
    new_time_ev = time_ev + abs(new_x - x)
    for t in range(math.ceil(time_ev), math.floor(new_time_ev) + 1):
        data.append(x + sigma * (t - time_ev))
    x = new_x
    time_ev = new_time_ev
    sigma *= -1

plt.title('Zig-Zag 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