1549380323-Statistical Mechanics Theory and Molecular Simulation

Transition path sampling 307

in Fig. 7.5, which shows a thermostatted trajectory of a particle in aone-dimensional
double-well potential with minima atx=±1 and a barrier height of 8kT. The figure
shows that actual crossing events are rapid but times between such events are quite
long.
In the transition path sampling method, we approach the problem in away that
is qualitatively different from what we have done until now. Let us assume that we
knowa priorithe regions of phase space in which the trajectory initiates and in which
it finally ends up. We denote these regions generically asAandB, respectively (see
Fig. 7.6). Let us also assume that a timeTis needed for the system to pass fromAtoB.

A

B

.

x 0

.

x

.

∆t

x(k-1)∆t

.xk∆t

.x(k+1)∆t.

. xn∆t

..

..

.

. ..

Fig. 7.6A representative transition path fromAtoB. Also shown are discrete phase space
points along the path: Initial (x 0 ), final (xn∆t= xt), and several intermediate (x(k−1)∆t, xk∆t,
x(k+1)∆t) points.

If we could generate a molecular dynamics trajectory fromAtoB, then this trajectory
would consist of a sequence of discrete phase space points x 0 ,x∆t,x2∆t,...,xn∆t, where
n∆t=T, such that x 0 ∈Aand xn∆t∈B(see Fig. 7.6). Let us denote this set of
phase space points as X(T). That is, X(T) is a time-ordered sequence of microscopic
states visited by the system as it passes fromAtoB. If we view X(T) as belonging to
an ensemble of trajectories fromAtoB, then we can derive a probabilityPAB[X(T)]
associated with a given trajectory. Note that we are using functional notation because
PABdepends on the entire trajectory. If we regard the sequence ofphase space points
x 0 ,x∆t,x2∆t,...,xn∆tas a Markov chain, then there exists a ruleT(x(k+1)∆t|xk∆t) for
generating x(k+1)∆tgiven xk∆t. For example, suppose we posit that the trajectory is to
be generated deterministically via a symplectic integrator such as the velocity Verlet
method of Sec. 3.10. Then, x(k+1)∆twould be generated from xk∆tusing a Trotter
factorization of the propagator

x(k+1)∆t= eiL^2 ∆t/^2 eiL^1 ∆teiL^2 ∆t/^2 xk∆t≡φ∆t(xk∆t). (7.7.1)

Here,φ∆t(xk∆t) is a shorthand notation for the Trotter factorized single-step propa-
gator acting on xk∆t. The ruleT(x(k+1)∆t|xk∆t) must specify that there is only one
possible choice for x(k+1)∆tgiven xk∆t, which means we must take the rule as

T(x(k+1)∆t|xk∆t) =δ

(

x(k+1)∆t−φ∆t(xk∆t)

)

. (7.7.2)

In an ensemble of trajectories X(T), the general statistical weightP[X(T)] that we
would assign to any single trajectory is