1549380323-Statistical Mechanics Theory and Molecular Simulation

Transition path sampling 309

generate a trajectory X(T) starting from Y(T). Both X(T) and Y(T) must be transi-
tion paths fromAtoB. The detailed balance condition appropriate for the transition
path ensemble is

RAB[X(T)|Y(T)]PAB[Y(T)] =RAB[Y(T)|X(T)]PAB[X(T)]. (7.7.7)

As we did in eqn. (7.3.25), we decomposeRAB[X(T)|Y(T)] into a product

RAB[X(T)|Y(T)] = ΛAB[X(T)|Y(T)]TAB[X(T)|Y(T)] (7.7.8)

of a trial probabilityTABand an acceptance probability ΛAB. The same logic used to
obtain eqn. (7.3.28) leads to the acceptance rule for transition path sampling

Λ[X(T)|Y(T)] = min

[

1 ,

PAB[X(T)]TAB[X(T)|Y(T)]

PAB[Y(T)]TAB[Y(T)|X(T)]

]

. (7.7.9)

Since the trajectory Y(T) as assumed to be a proper transition path fromAtoB,
hA(y 0 ) = 1 andhB(yn∆t) = 1. Thus, we can write eqn. (7.7.9) as

Λ[X(T)|Y(T)] =hA(x 0 )hB(xn∆t)min

[

1 ,

P[X(T)]TAB[X(T)|Y(T)]

P[Y(T)]TAB[Y(T)|X(T)]

]

, (7.7.10)

which is zero unless the new trajectory X(T) is also a proper transition path.
As with any Monte Carlo algorithm, the key to efficient sampling of transition paths
is the design of the ruleTAB[X(T)|Y(T)] for generating trial moves from one path to
another. Here, we will discuss a particular type of trial move knownas a “shooting
move.” Shooting moves are conceptually simple. We randomly select a point yj∆t(by
randomly choosing the integerj) from the starting trajectory Y(T) and modify it in
some way to give a point xj∆ton the trial trajectory X(T), referred to as a “shooting
point.” Starting from this point, trajectories are launched forward and backward in
time. If the new trajectory X(T) thus generated is a transition path fromAtoB, it is
accepted with some probability; otherwise, it is rejected. The idea of shooting moves
is illustrated in Fig. 7.7.
Letτ(xj∆t|yj∆t) denote the rule for generating a trial shooting point xj∆tfrom
yj∆t. Then, we can expressTAB[X(T)|Y(T)] as

TAB[X(T)|Y(T)] =

τ(xj∆t|yj∆t)





n∏− 1

k=j

T(x(k+1)∆t|xk∆t)





[ j

∏

k=1

T(x(k−1)∆t|xk∆t)

]

. (7.7.11)

The first product in eqn. (7.7.11) is the probability for the forward trajectory, and
the second product, which requires the rule to generate x(k−1)∆tfrom xk∆tvia time-
reversed dynamics, is the weight for the backward trajectory. For molecular dynamics,
this part of the trajectory is just obtained via eqn. (7.7.1) using−∆tinstead of ∆t,
i.e., x(k−1)∆t=φ−∆t(xk∆t), which can be generated by integrating forward in time