ZigZag X2X4.py

From Werner KRAUTH

Contents 1 Context 2 Python program 3 Further information 4 References

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