10.3 Importance sampling through Markov chains 307
0 20 40 60 80 100 120
Magnetisation
10 MCS^3
–1
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1
Figure 10.2. Magnetisation of a 20×20 Ising lattice as a function of time for
an effective coupling constantJ/kBT=0.46. The graph clearly shows plateaux
where the magnetisation fluctuates around positive or negative average values.
- Making a histogram of the magnetisation as mentioned above and taking the
peak positions as the value of the magnetisation. Again, distinguishing the peak
will be difficult when critical fluctuations are strong.
It will be clear that measuring the magnetisation close to the phase transition is a
difficult task which should be avoided if possible.
It should be noticed that the Ising model is formulated without a prescription
for its dynamical evolution. Consequently, there exists no molecular dynamics
method for the Ising model, and Monte Carlo is the only simulation technique
for spin systems such as the Ising model. Using the Metropolis method, the Ising
model becomes dynamic in the sense that its configurations change with time. The
kinetics of this behaviour have been studied, first because it is assumed that the
dynamic evolution of Ising-like systems in nature is governed by processes not too
dissimilar from Metropolis MC and second because the kinetics are relevant to the
reliability of the simulations, in particular near the critical point [ 13 , 14 ]. Recall
that the dynamical behaviour close to the phase transition is expressed in terms of
the dynamic critical exponent,z, which describes the divergence of the correlation
time (seeSection 7.3.2).
Our choice for the matrixωXX′is not the only possibility. We may allow for two
(or more) spins flipping over at the same step, and these spins might be restricted
to be opposite so that the total magnetisation does not change, giving us a constant-
magnetisation algorithm. In all cases, instead of selecting the spins randomly, they
may be chosen in a regular fashion, for example by scanning through the lattice row
by row. In that case, one step in the Markov chain consists of a scan through the
whole lattice.ωX,X′is equal foranynew configurationX′and it remains to show
