03 -
0
02 A
position of spin si
10 20 30 40 50 60 70 80 90
Figure 4-2. Test results for ID Ising system. The graph shows the correlation C between
spins si and S20 so that the y-axis shows: C(i) = (sis20) and the x-axis labels
the spin index i. There are three sets of data plotted together, one data-set
for each value of the Ising-coupling g. The graph also depicts the exponential
fits done so that the exponential falloff of the correlation can be read off. The
results are systematically organized in a table below.
these simple tests, or examples of application, we can allow ourselves to spend some time
to prepare for the real application of the Monte Carlo method presented in this thesis.
4.1.5 Statistical Errors and Data Analysis
Being a stochastic numerical method, the Monte Carlo approach gives only
statistically significant results. We saw this clearly in the last section where even in
the very simple cases of a bosonic chain and a ID Ising spin system, where deterministic
methods would probably have served better (in fact, exact answers were available!). One
might think that since the Monte Carlo method indeed performed relatively poorly in the
simplest cases, it is likely to fail utterly when a more complicated system is considered.
This however is by no means the case. The statistical inaccuracies of the Monte Carlo
method are inherent in the algorithm and remain in the more complicated applications,
however, there is nothing which indicates that this effect should increase in any way just
because the system under consideration is complicated. To understand this, we will discuss
briefly some of the statistical observations which are standard in the application of Monte
Carlo methods [19, 20]. (The book by Lyons [21] contains many useful discussions on
statistics in general scenarios.) Although structurally this section belongs here with the