Computational Physics - Department of Physics

446 13 Monte Carlo Methods in Statistical Physics

ter 11. There we considered a box divided into two equal halves separated by a wall. At the
beginning, timet= 0 , there areNparticles on the left side. A small hole in the wall is then
opened and one particle can pass through the hole per unit time. After some time the system
reaches its equilibrium state with equally many particles in both halves,N/ 2. Thereafter, the
mean number of particles oscillates aroundN/ 2.
The number of Monte Carlo cycles needed to reach this equilibrium position is referred to
as the thermalization time, or equilibration timeteq. We should then discard the contributions
to various expectation values till we have reached equilibrium. How to determine the ther-
malization time can be done in a brute force way, as demonstrated in Figs. 13.9 and 13.10.
In Fig. 13.9 the calculations have been performed with a 40 × 40 lattice for a temperature
kBT/J= 2. 4 , which corresponds to a case close to a disordered system. Wecompute the abso-
lute value of the magnetization after each sweep over the lattice. Two starting configurations
were used, one with a random orientation of the spins and one with an ordered orienta-
tion, the latter corresponding to the ground state of the system. As expected, a disordered
configuration as start configuration brings us closer to the average value at the given temper-
ature, while more cycles are needed to reach the steady statewith an ordered configuration.
Guided by the eye, we could obviously make such plots and discard a given number of sam-
ples. However, such a rough guide hides several interestingfeatures. Before we switch to a
more detailed analysis, let us also study a case where we start with the ’correct’ configuration
for the relevant temperature.

Random start configuration

Ground state as start

t

|M

|

0 1000 2000 3000 4000 5000

1

0.8

0.6

0.4

0.2

0

Fig. 13.9Absolute value of the mean magnetisation as function of timet. Time is represented by the number
of Monte Carlo cycles. The calculations have been performedwith a 40 × 40 lattice for a temperaturekBT/J=
2. 4. Two start configurations were used, one with a random orientation of the spins and one with an ordered
orientation, which corresponds to the ground state of the system.

Fig. 13.10 displays the absolute value of the mean magnetisation as function of timetfor a
100 × 100 lattice for temperatureskBT/J= 1. 5 andkBT/J= 2. 4. For the lowest temperature,
an ordered start configuration was chosen, while for the temperature close to the critical
temperature, a disordered configuration was used. We noticethat for the low temperature
case the system reaches rather quickly the expected value, while for