486 The Feynman path integral
acceptance probability is 40%. For a system ofNparticles, the algorithm extends
straightforwardly. First a cyclic polymer is chosen at random, and aMonte Carlo pass
is performed on that chain. On average,Nsuch passes will move the entire system.
We conclude this subsection by noting an important difference between path-
integral Monte Carlo and path-integral molecular dynamics. In molecular dynamics,
a single time step generates a move of the entire system, while in Monte Carlo, each
individual attempted move only changes a part of the system. This difference becomes
important when implementing path integrals on parallel computing platforms. Path-
integral molecular dynamics parallelizes much more readily than staging or normal-
mode path-integral Monte Carlo. Thus, if a molecular dynamics algorithm could be
constructed with a convergence efficiency that rivals Monte Carlo,then the former
becomes a competitive method. As part of our discussion of thermodynamic estimators
in the next subsection, we will also present a comparison of the molecular dynamics
and Monte Carlo approaches for a simple system.
12.6.3 Numerical aspects of thermodynamic estimators
As noted in Section 12.3, the estimators in eqns. (12.3.20), (12.3.24), and (12.5.12)
suffer from large fluctuations in the kinetic energy due to their lineardependence on
P. The consequence of this dependence is that in highly quantum systems, which
require a large number of discretizations, it becomes increasingly difficult to converge
such estimators. A solution to this dilemma was presented by Herman, Bruskin, and
Berne (1982), who employed a path integral version of the virial theorem. For a single
particle in one dimension, the theorem states
P
2 β−
〈
1
2
mωP^2∑P
k=1(xk−xk+1)^2〉
f=
〈
1
2 P
∑P
k=1xk∂U
∂xk〉
f. (12.6.33)
Before we prove this theorem, we demonstrate its advantage in a simple application.
First, note that when eqn. (12.6.33) is substituted into eqn. (12.3.19), a new energy
estimator known as thevirial energy estimatorresults:
εvir(x 1 ,...,xP) =1
P
∑P
k=1[
1
2
xk∂U
∂xk+U(xk)]
. (12.6.34)
The elimination of the kinetic energy yields an energy estimator with a much lower
variance and, therefore, better convergence behavior than the primitive estimator of
eqn. (12.3.20). In Fig. 12.12, we show a comparison between the instantaneous fluctu-
ations and cumulative averages of the primitive and virial estimatorsfor a harmonic
oscillator withm= 1 andω= 10 computed using staging molecular dynamics with
P= 32,P= 64, andP= 128 beads. The figure shows how the fluctuations, shown
in grey, grow withP while the fluctuations in the virial estimator are insensitive to
P. Despite the fact that the fluctuations of the primitive estimator grow withP, the
cumulative averages between the two estimators agree for allP. This illustrates the
idea that in any path-integral simulation, one should monitor both estimators and
ensure that they agree. If they do not, this should be taken as a sign of a problem in
the simulation.