1549380323-Statistical Mechanics Theory and Molecular Simulation

Path sampling 597

where the displacementδxdis purely deterministic, andδxris due to the random force.
If we take the random force to be a Gaussian random variable, thenwe can state the
rule for generating trial moves in phase space from xk∆tto x(k+1)∆tas

T(x(k+1)∆t|xk∆t) =w(δxr), (15.6.20)

wherew(x) is a Gaussian distribution of width determined by the friction (Chan-
drasekhar, 1943). In the high friction limit, eqn. (15.6.20) can be shown to be

T(q((k+ 1)∆t)|q(k∆t)) =

√

μγ

4 πkT∆t

exp

[

−

μγ

4 kT∆t

(q((k+ 1)∆t)−q(k∆t))^2

]

(15.6.21)

for a single degree of freedomq (Dellagoet al., 2002). Note that as ∆t→0, the
Gaussian distribution tends to a Diracδ-function, as expected (see eqn. (A.5) in Ap-
pendix A).
The second difference in the shooting algorithm is in the choice of the shooting
point. Because the trajectories are stochastic, we are free to choose a shooting point
to lie on the old trajectory Y(t) without modification because a trajectory launched
from this point will be different from Y(t). Thus, given a stochastic transition path
Y(t) and a randomly chosen point yj∆ton this path, we can take the rule for generating
the new shooting point xj∆tto be

τ(xj∆t|yj∆t) =δ(xj∆t−yj∆t). (15.6.22)

Third, because the Langevin equation acts as a thermostatting mechanism, the distri-
butions of initial conditionsf(x 0 ) andf(y 0 ) will be canonical by construction. Thus,
if canonical sampling is sought, then there is no need to apply the acceptance rule
min[1,f(x 0 )/f(y 0 )] for each trial path move. Putting this fact together with eqn.
(15.6.22) and the Gaussian form of eqn. (15.6.20), which is symmetric, gives a partic-
ularly simple acceptance criterion from eqn. (7.7.12)

Λ[X(t)|Y(t)] =hA(x 0 )hB(xn∆t). (15.6.23)

Thus, as long as the new path is a proper transition path fromAtoB, it is accepted
with probability 1. We can now summarize the steps of the shooting algorithm for
stochastic paths as follows:

1. Choose an indexjrandomly on the old trajectory Y(t) and take the shooting
   point yj∆tto be the shooting point xj∆t.


2. Integrate the equations of motion backwards in time from the shooting point to
   the initial condition x 0 using a stochastic propagation scheme such as that of eqn.
   (15.5.17).


3. If the initial condition x 0 is not in the phase space regionA, reject the trial move;
   otherwise, accept it.


4. Integrate the equations of motion forward in time to generate the final point xn∆t
   using a stochastic propagation scheme such as that of eqn. (15.5.17).