1549380323-Statistical Mechanics Theory and Molecular Simulation

308 Monte Carlo

P[X(T)] =f(x 0 )

n∏− 1

k=0

T(x(k+1)∆t|xk∆t), (7.7.3)

wheref(x 0 ) is the equilibrium distribution of initial conditions x 0 , for example, a
canonical distributionf(x 0 ) = exp(−βH(x 0 ))/Q(N,V,T). However, since our interest
is in trajectories that start in the phase space regionAand end in the regionB, we
need to restrict the trajectory distribution in eqn. (7.7.3) to this subset of trajectories.
We do this by multiplying eqn. (7.7.3) by functionshA(x 0 ) andhB(xn∆t), where
hA(x) = 1 if x∈AandhA(x) = 0 otherwise, with a similar definition forhB(x). This
gives the transition path probabilityPAB[X(T)] as

PAB[X(T)] =

1

FAB(T)

hA(x 0 )P[X(T)]hB(xn∆t), (7.7.4)

whereFABis a normalization constant given by

FAB(T) =

∫

dx 0 ···dxn∆thA(x 0 )P[X(T)]hB(xn∆t). (7.7.5)

Eqns. (7.7.4) and (7.7.5) can be regarded as the probability distribution and partition
function for an ensemble of trajectories that begin inAand end inBand thus, they
can be regarded as defining an ensemble called thetransition path ensemble(Dellago
et al., 1998; Bolhuiset al., 2002; Dellagoet al., 2002). Although eqns. (7.7.4) and (7.7.5)
are valid for any trajectory ruleT(x(k+1)∆t|xk∆t), if we take the specific example of
deterministic molecular dynamics in eqn. (7.7.2), then eqn. (7.7.5) becomes

FAB(T) =

∫

dx 0 ···dxn∆thA(x 0 )f(x 0 )

n∏− 1

k=0

δ

(

x(k+1)∆t−φ∆t(xk∆t)

)

hB(xn∆t)

=

∫

dx 0 f(x 0 )hA(x 0 )hB(xn∆t(x 0 )), (7.7.6)

where we have used the Diracδ-functions to integrate over all points xk∆texcept
k= 0. When this is done thehBfactor looks likehB(φ∆t(φ∆t(···φ∆t(x 0 )))), where
the integratorφ∆t actsntimes on the initial condition x 0 to give the unique nu-
merical solution xn∆t(x 0 ) appearing in eqn. (7.7.6). Thus, for deterministic molecular
dynamics, eqn. (7.7.6) simply counts those microstates belonging tothe equilibrium
ensemblef(x 0 ) that are contained inAand, when integrated fornsteps, end inB. In
Chapter 15, we will show how to define and generate the transition path ensemble for
trajectories obeying stochastic rather than deterministic dynamics.
Having now defined the ensemble of transition paths, we need a Monte Carlo algo-
rithm for sampling this ensemble. The method we will describe here is anadaptation
of the M(RT)^2 algorithm for an ensemble of paths rather than one of configurations.
Accordingly, we seek to generate a Markov chain ofMtrajectories X 1 (T),...,XM(T),
and to accomplish this, we begin, as we did in Section 7.3.3, with a generalization of
the detailed balance condition. LetRAB[X(T)|Y(T)] be the conditional probability to