Computational Physics

10.6 Further applications and Monte Carlo methods 325

In this example that constant should be near 1/(0.75Nθ), whereNθis the number
of choices for the angleθ(6 in the example).

programming exercise
You can now try to code the PERM algorithm. If you have done it correctly
you should be able to reproduce the double crosses inFigure 10.3. As you
can see, they fall on top of the theoretical fit which hasν= 3 /4.
It should be mentioned that there exist many more methods than described in this
subsection. In particular we mention the configurational bias Monte Carlo (CBMC)
method by[43].

10.6.2 Tempering and replica exchange

If we want to simulate a disordered system at low temperatures, we run into the
problem that many low-energy states exist. Once the system finds itself in phase
space near such a low-energy state, it can escape only with great difficulty: the
system is trapped. In order to sample the phase space, we must visit a large set of
minima accessible at the relevant temperature, and therefore we should somehow
‘push’ the system over the barriers separating the minima. The problem which arises
is that pushing the system over the barrier is likely to destroy the detailed balance
condition. Methods preserving detailed balance and moving the system efficiently
through phase space are thesimulated tempering Monte Carlo[44], thereplica
Monte Carlo[45, 46] and thereplica exchange[47]method. All these methods
are based on the idea that the system can move from one minimum to another by
repetitive heating and cooling. The problem is to heat and cool efficiently such
that moves to different temperatures have a reasonable acceptance probability and
that overall detailed balance is preserved. We briefly describe the replica exchange
method in this section.
Mreplicas of the system under consideration are simulated in parallel, each at
a different (inverse) temperatureβ 1 ,β 2 ,...,βM. We takeβ 1 <β 2 <...<βM.
We now alternate a number of ordinary MC steps for each of the replicas by an
exchange of the replica configurations, or, equivalently, of the replica temperat-
ures. Exchanging the temperatures just involves assigning different numbers to the
temperatures in the two replicas but the temperatures are then no longer ordered.
Exchanging the configurations will be more time-consuming.
The partition function describing the collection of replicas is given as

Z=

∏M

m= 1

Z(βm)=Z(β 1 )Z(β 2 )...Z(βM). (10.54)