Levy harmonic.py
From Werner KRAUTH
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| - | This page is part of my public lectures on [https://www.physics.ox.ac.uk/events/algorithms-and-computations-theoretical-physics-set-lectures Algorithms and Computations in theoretical physics] at the University of Oxford (see the announcement for a syllabus). For more information, see [[Oxford_Lectures_2025|the main page of the public lectures]]. It presents the Lévy construction, in other words the direct sampling algorithm invented by P. Lévy in 1940. For an in-depth discussion, see my below reference. Other programs in the same series are the Metropolis algorithm and the Hamiltonian Monte Carlo algorithm. | + | This page is part of my public lectures on [https://www.physics.ox.ac.uk/events/algorithms-and-computations-theoretical-physics-set-lectures Algorithms and Computations in theoretical physics] at the University of Oxford (see the announcement for a syllabus). For more information, see [[Oxford_Lectures_2025|the main page of the public lectures]]. It presents the Lévy construction, in other words the direct sampling algorithm invented by P. Lévy in 1940. For an in-depth discussion, see my below reference. Other programs in the same series are the [[Metropolis_harmonic.py|Metropolis algorithm]] and the [[HMC_harmonic.py|Hamiltonian Monte Carlo algorithm]], both for the harmonic chain. |
| ==Python program== | ==Python program== | ||
Revision as of 19:07, 19 February 2025
Contents |
Context
This page is part of my public lectures on Algorithms and Computations in theoretical physics at the University of Oxford (see the announcement for a syllabus). For more information, see the main page of the public lectures. It presents the Lévy construction, in other words the direct sampling algorithm invented by P. Lévy in 1940. For an in-depth discussion, see my below reference. Other programs in the same series are the Metropolis algorithm and the Hamiltonian Monte Carlo algorithm, both for the harmonic chain.
Python program
import math, random
def U(x):
U = 0.0
for k in range(N):
k_minus = (k - 1) % N
x_minus = x[k_minus]
if k == 0: x_minus -= L
U += (x[k] - x_minus) ** 2 / 2.0
return U
N = 8
L = 16
delta = 1.0
Iter = 1000000
for iter in range(Iter):
xstart = random.uniform(0.0, L)
xend = xstart + L
x = [xstart]
for k in range(1, N): # loop over internal slices
x_mean = ((N - k) * x[k - 1] + xend) / (1.0 + N - k)
sigma = math.sqrt(1.0 / (1.0 + 1.0 / (N - k) ))
x.append(random.gauss(x_mean, sigma))
x.append(xend)
Further information
References
Krauth, W. Hamiltonian Monte Carlo vs. event-chain Monte Carlo: an appraisal of sampling strategies beyond the diffusive regime. ArXiv: 2411.11690
