1549380323-Statistical Mechanics Theory and Molecular Simulation

310 Monte Carlo

A

B

.yj∆t

.

xj∆t

.

xj∆t

Accepted

Rejected

Fig. 7.7The shooting algorithm. The original path Y(T) is shown as the solid line. A point
yj∆trandomly chosen from this path is used to determine the shooting point xj∆t. Two
example shooting paths are shown as dashed lines. The long dashed line is a successful path
that is accepted, while the short dashed line shows an unsuccessful path that is rejected.

with time step ∆tbut with velocities reversed at the shooting point. Combining eqn.
(7.7.11) with (7.7.10), we obtain for the acceptance probability

Λ[X(T)|Y(T)] =hA(x 0 )hB(xn∆t)min

[

1 ,

f(x 0 )

f(y 0 )

(n− 1

∏

k=0

T(x(k+1)∆t|xk∆t)

T(y(k+1)∆t|yk∆t)

)

×



τ(yj∆t|xj∆t)

τ(xj∆t|yj∆t)

n∏− 1

k=j

T(y(k+1)∆t|yk∆t)

T(x(k+1)∆t|xk∆t)

j∏− 1

k=0

T(yk∆t|y(k+1)∆t)

T(xk∆t|x(k+1)∆t)









=hA(x 0 )hB(xn∆t)

×min

[

1 ,

f(x 0 )

f(y 0 )

τ(yj∆t|xj∆t)

τ(xj∆t|yj∆t)

j∏− 1

k=0

T(yk∆t|y(k+1)∆t)T(x(k+1)∆t|xk∆t)

T(xk∆t|x(k+1)∆t)T(y(k+1)∆t|yk∆t)

]

. (7.7.12)

Although eqn. (7.7.12) might seem rather involved, consider what happens when the
trajectories are generated by molecular dynamics, with the trial probability given by
eqn. (7.7.2). Since a symmetric Trotter factorization of the classical propagator is time
reversible, as discussed in Section 3.10, the ratioT(yk∆t|ry(k+1)∆t)/T(y(k+1)∆t|yk∆tis
unity, as is the ratioT(x(k+1)∆t|xk∆t)/T(xk∆t|x(k+1)∆t), and the acceptance criterion
simplifies to

Λ[X(T)|Y(T)] =hA(x 0 )hB(xn∆t)min

[

1 ,

f(x 0 )

f(y 0 )

τ(yj∆t|xj∆t)

τ(xj∆t|yj∆t)

]

. (7.7.13)

Finally, suppose the new shooting point xj∆tis generated from the old point yj∆t
using the following rule;
xj∆t= yj∆t+ ∆, (7.7.14)