306 Monte Carlo
aim hasonlybeen to sample, there is no dynamical information in the sequence of
configurations we have produced. Consequently, these methodscan only be used to
calculate equilibrium and thermodynamic properties. But suppose wecould devise a
Monte Carlo scheme capable of producing actual dynamical trajectories from one part
of phase space to another. The framework of Monte Carlo is flexibleenough that such
a thing ought to be possible. In fact, this is something that can be achieved, but doing
so requires a small shift in the way we think of statistical ensembles and the sampling
problem.
The technique we will discuss, known astransition path sampling, was pioneered
by Chandler and coworkers (Dellagoet al., 1998; Bolhuiset al., 2002; Dellagoet al.,
2002; Frenkel and Smit, 2002). This approach is a particularly powerful one for gener-
ating dynamical trajectories between two regions of phase spacewhen the passage of
the system from one to the other is exceedingly rare. When suchrare events(which
we will discuss in more detail in Section 8.5) cannot be accessed using ordinary molec-
ular dynamics, transition path sampling provides a means of producing the desired
trajectories. A classic example of a rare event is the dissociation ofa water molecule
according to 2H 2 O(l)−→H 3 O+(aq) + OH−(aq) (Geissleret al., 2001). The reaction
ostensibly only requires transferring a proton from one water molecule to another.
However, if we attach ourselves to a particular water molecule and wait for the chem-
ical reaction to occur, the average time we would have to wait is ten hours for a single
event, a time scale that is well beyond that which can be accessed in an ordinary
molecular dynamics calculation. Generally, a process is termed a rare-event process
when the system must cross one or more high energy barriers (seeFig. 7.2). The ac-
tual passage time over the barrier can be quite short, while most ofthe time is spent
waiting for a sufficient amount of energy to amass in a small number ofmodes in the
system to allow the barrier to be surmounted. When a system is in equilibrium, where
equipartitioning holds, such a fluctuation is, indeed, a rare event. This is illustrated
0 2000 4000 6000 8000 10000
t
-2
-1
0
1
2
x
Fig. 7.5 Rare-event trajectory in a double-well potential with barrier height 8kT.