1549380323-Statistical Mechanics Theory and Molecular Simulation

484 The Feynman path integral

12.6.2 Path-integral Monte Carlo

We saw in Chapter 7 that Monte Carlo methods are very effective forsampling an
equilibrium distribution such as the canonical ensemble. Therefore,it is worth using a
little space to discuss the calculation of path integrals using a Monte Carlo approach.
The algorithm we will describe here uses many of the same ideas discussed in the previ-
ous subsection. In particular, for a quantum free particle in one dimension (U(ˆx) = 0),
the discretized action expressed in terms of staging or normal-mode variables is just
a sum of uncoupled harmonic oscillators, and as we saw in Section 3.8.3,these can be
easily sampled using the Box-Muller method. The idea of path-integral Monte Carlo,
then, is to construct an M(RT)^2 algorithm (see Section 7.3.3), in which we sample
the free particle distribution directly and use the change in the potential energy to
build an acceptance probability. However, unlike path-integral molecular dynamics,
where a staging or normal-mode transformation can be applied to the entire cyclic
polymer chain, the same cannot always be done in path-integral Monte Carlo. The
reason for this is that ifPis sufficiently large, the complete set of staging or normal
modes is simply too large to be sampled in its entirety in one Monte Carlo move: the
average acceptance probability would, for most problems, simply betoo low. Thus,
in path-integral Monte Carlo, staging or normal-mode transformations are applied to
segments of the cyclic polymer chain of a certain lengthjthat must be optimized to
give a desired average acceptance probability.
In fact, we have already seen how to perform both normal-mode and staging trans-
formations to a set ofjparticles with a harmonic nearest-neighbor coupling anchored
to fixed endpoints in Sections 1.7 and 4.5, respectively. We first describe the staging
transformation. The idea of staging was originally introduced by Ceperley and Pol-
lock (1984) as a means of constructing an efficient Monte Carlo scheme. However,
explicit variable transformations were not employed in the original work. Here we
modify the original staging algorithm to incorporate explicit transformations. In order
to sample a segment of lengthjof the free particle distribution for the cyclic poly-
mer chain, we start by randomly choosing a starting bead. Supposethat the chosen
bead has an imaginary-time indexl. This bead forms one of the fixed endpoints of
the segment and the other isl+j+ 1 beads away from this one. This leaves us with
jintermediate beads having primitive coordinatesxl+1,...,xl+j. We now transform
these primitive variables to staging variables using eqn. (4.5.38), which, for this case,
appears as follows:

ul+k=xl+k−

kxl+k+1−xl

(k+ 1)

k= 1,...,j. (12.6.26)

Eqn. (4.5.39) also allows a recursive inverse to be defined as

xl+k=ul+k+

k

k+ 1

xl+k+1+

1

k+ 1

xl. (12.6.27)

Applying the transformation to the following portion of the quantumkinetic energy

1

2

mω^2 P

∑j

k=0

(xl+k−xl+k+1)^2 ,