Replica exchange 301Clearly, a straightforward molecular dynamics or Monte Carlo calculation car-
ried out on a rough potential energy surface exhibits hopelessly slow convergence of
equilibrium properties because the probability of crossing a barrier of heightU‡is
proportional to exp(−βU‡). Thus, the system tends to become trapped in a single
local minimum and requires an enormously long time to escape the minimum. As an
illustration, the Boltzmann factor for a barrier of height 15 kJ/molat a temperature of
300 K is roughly 3× 10 −^3 , and for a barrier height of 30 kJ/mol, it is roughly 6× 10 −^6.
As a result, barrier crossing becomes a “rare event.” In Chapter 8, we will discuss a
number of methods for addressing the rare-event problem. Here, we begin studying
this problem by introducing a powerful and popular method, replica exchange Monte
Carlo, designed to accelerate barrier crossing.
The term “replica exchange” refers to a class of Monte Carlo methods in which
simultaneous calculations are performed on a set ofMindependent copies or replicas
of a physical system. Each replica is assigned a different value of some physical control
parameter, and Monte Carlo moves in the form of exchanges of thecoordinates between
different replicas are attempted. In this section, we will describe a replica exchange
approach calledparallel tempering(Marinari and Parisi, 1992; Tesiet al., 1996), in
which temperature is used as the control variable, and different temperatures are
assigned to the replicas.
In the parallel tempering scheme, a set of temperaturesT 1 ,...,TM, withTM >
TM− 1 >···> T 1 is selected and assigned to theMreplicas. The lowest tempera-
tureT 1 is taken to be the temperatureTof the canonical distribution to be sampled.
The motivation for this scheme is that the high-temperature replicas can easily cross
barriers on the potential energy surface if the temperatures are high enough. The at-
tempted exchanges between the replicas cause the coordinates of the high-temperature
copies to “percolate” down to the low-temperature copies, allowingthe latter to sam-
ple larger portions of the configuration space at the correct temperature. The idea
is illustrated in Fig. 7.3. On the rough one-dimensional surface shownin the figure,
the low-temperature copies sample the lowest energy minima on the surface, while the
high-temperature copies are able to “scan” the entire surface.
Letr(1),...,r(M) be the complete configurations of theM replicas, i.e.r(K) ≡
r( 1 K),...,r(NK). Since the replicas are independent, the total probability distribution
F(r(1),...,r(K)) for the full set of replicas is just a product of the individual distribution
functions of the replicas
F(r(1),...,r(K)) =∏M
K=1fK(
r(K))
(7.5.1)
and is, therefore, separable. Here,
fK(
r(K))
=
exp[
−βKU(
r(K))]
Q(N,V,TK)
. (7.5.2)
A replica exchange calculation proceeds by performing either a molecular dynamics
or simple M(RT)^2 Monte Carlo calculation on each individual replica. Periodically,
a neighboring pair of replicasK andK+ 1 is selected, and an attempted move