From Werner KRAUTH

This page presents the Python3 program markov_ising.py, a Markov-chain algorithm for the Ising model on an LXL square lattice in two dimensions, with periodic boundary conditions. The program uses the Metropolis algorithm.

Description

The program is described in my 2024 Oxford Lecture No 8, and also in my book. This version of the program estimates the energy per particle, and the specific heat.

Program

import random, math

L = 6
N = L * L
nbr = {i : ((i // L) * L + (i + 1) % L, (i + L) % N,
            (i // L) * L + (i - 1) % L, (i - L) % N) \
                                    for i in range(N)}
nsteps = 10000000
T = 1.0
beta = 1.0 / T
S = [random.choice([1, -1]) for k in range(N)]
E = -0.5 * sum(S[k] * sum(S[nn] for nn in nbr[k]) \
                                for k in range(N))
E_tot, E2_tot = 0.0, 0.0
for step in range(nsteps):
    k = random.randint(0, N - 1)
    h = sum(S[nn] for nn in nbr[k])
    Sk_old = S[k]
    delta_E = 2.0 * S[k] * h
    if random.uniform(0.0, 1.0) < math.exp(-beta * delta_E):
        S[k] *= -1
        E -= 2.0 * h * S[k]
    E_tot += E
    E2_tot += E ** 2
E_av  = E_tot / float(nsteps)
E2_av = E2_tot / float(nsteps)
c_V = beta ** 2 * (E2_av - E_av ** 2) / float(N)
print(E_av / N, c_V)

Version

See history for version information.