Computational Physics

326 The Monte Carlo method

This can also be written as

Z=

∑

Xm;m=1,...,M

exp

(

−

∑M

m= 1

βmEXm

)

, (10.55)

whereXmdenotes a system configuration of replicam. We can interpret this partition
function as that of a large system, encompassing all replicas, with a ‘Boltzmann
factor’

P(β 1 ,X 1 ;...;βM,XM)=exp

(

−

∑M

m= 1

βmEXm

)

. (10.56)

Note that theβmare stochastic variables in this formulation. They can exchange
their values but are subject to the condition that the set of values taken on by
theβmremains the same. For this ensemble we perform a Metropolis Monte Carlo
algorithm. Suppose we let the replicas evolve independently according to the stand-
ard Metropolis algorithm. Then obviously detailed balance is satisfied. After a
number of steps, however, we exchange the configurations (or the temperatures)
pair-wise according to

βn,Xn;βm,Xm→βn,Xm;βm,Xn. (10.57)

This means that a low-temperature replica receives a high(er) temperature configur-
ation. At higher temperatures, in particular above the phase transition temperature,
the system moves easily over all the free energy barriers in phase space. If the low-
temperature replicas move to such high temperatures and back, their configuration
will in general have moved to different (free) energy minima.
We calculate the expectation values of physical quantitiesAat any of the tem-
peratures by averaging over all replica configurations at that particular temperature
(perhaps omitting the first few MCS just after a temperature change). Note that
exchanges are performed only between adjacent temperatures (βmandβm+ 1 if
the temperatures are ordered), as the acceptance rate decreases exponentially with
temperature difference.
Let us calculate the transition probability for temperature exchange. The ratio
between a forward and a backward move is given by

T(βn,Xn;βm,Xm→βn,Xm;βm,Xn)

T(βn,Xm;βm,Xn→βn,Xn;βm,Xm)

=exp[−βm(EXn−EXm)−βn(EXn−EXm)]

=exp[−(βm−βn)(EXn−EXm)]. (10.58)

This implies that a trial step in which the temperatures are exchanged is accepted
with a probability min{1, exp[−(βm−βn)(EXn−EXm)]}.
For the method to be successful it is necessary that all Metropolis steps have a
reasonable acceptance. It is clear that the acceptance rate is sensitive to the differ-
ence between adjacent temperatures. For the acceptance rates to be of order 1, the