294 Monte Carlo
As a final observation, it is interesting to compare Monte Carlo with molecular
dynamics as methods for sampling ensemble distributions. A key difference, readily
apparent from the preceding discussion, is that in a Monte Carlo calculation, parti-
cles are moved one at a time (or, at most, a few at a time); in moleculardynamics,
all of the particles are moved simultaneously. It might seem, therefore, that molecular
dynamics calculations are more efficient than Monte Carlo, but if we recall that molec-
ular dynamics moves are limited by the size of the time step, we find that, in many
instances, the methods are comparable in their efficiency when molecular dynamics is
used with appropriate thermostatting and/or barostatting schemes. An advantage of
molecular dynamics over Monte Carlo is that it is straightforward to couple and un-
couple thermostats and barostats in order to switch between sampling and dynamics
calculations, which makes writing an elegant, object-oriented codethat encompasses
both types of calculations conceptually seamless. Monte Carlo, as described here, is
only useful as a sampling technique, and therefore a separate molecular dynamics
module would be needed to study the dynamics of a system. Becausemolecular dy-
namics moves all particles simultaneously, it is also easier to devise andimplement
algorithms suitable for parallel computing architectures in order totackle very large-
scale applications. On the other hand, Monte Carlo allows for considerable flexibility
to invent new types of moves since one need worry only about satisfying detailed bal-
ance. It is, of course, likewise possible to devise clever molecular dynamics methods,
as we have seen in Chapters 4 and 5. However, in molecular dynamics,“cleverness”
appears in the equations of motion and the demonstration that thealgorithm achieves
its objective. Finally, due to inherent randomness, Monte Carlo calculations are, by
construction, ergodic, even if a large number of Monte Carlo passes is required to
achieve converged results. In molecular dynamics, because of its deterministic nature,
achieving ergodicity is a significant challenge.
Before considering other ensembles, it is worth mentioning how the algorithm in
eqn. (7.3.37) is modified for systems consisting of rigid molecules. Since a rigid body
has both translational and rotational degrees of freedom (see Section 1.11), two types
of uniform moves are needed. Suppose a rigid body has a center-of-mass positionR
andnconstituent particles with coordinatesr 1 ,...,rnrelativeto the center-of-mass.
In a system consisting ofNsuch rigid bodies, we first choose one of them at random.
Then, eqn. (7.3.37) is applied to the center-of-mass in order to generate a move from
RtoR′according to:
X′=X+
1
√
3
(ξx− 0 .5) ∆Y′=Y+
1
√
3
(ξy− 0 .5) ∆Z′=Z+
1
√
3
(ξz− 0 .5) ∆. (7.3.38)Eqn. (7.3.38) generates a translation of the rigid body. Next, a unitvectornis ran-
domly chosen to define an axis through the center-of-mass. This can be accomplished
by choosing three additional random numbersζx,ζy, andζzto give the components