1549380323-Statistical Mechanics Theory and Molecular Simulation

Transition path sampling 311

where the phase space displacement ∆ is chosen randomly from a symmetric distribu-
tionπ(∆) satisfyingπ(−∆) =π(∆). In this case, the ratioτ(yj∆t|xj∆t)/τ(xj∆t|yj∆t)
in eqn. (7.7.13) is unity, and the acceptance rule becomes simply

Λ[X(T)|Y(T)] =hA(x 0 )hB(xn∆t)min

[

1 ,

f(x 0 )

f(y 0 )

]

, (7.7.15)

which is determined just by the initial conditions and whether the newtrajectory is a
proper transition path fromAtoB.
At this point, several comments on the shooting algorithm are in order. First, a very
common and simple choice for the phase space displacement is ∆ = (0,δp), meaning
that only the momenta are altered, while the configuration is left unchanged (Dellago
et al., 2002). Ifδpis chosen from a Maxwell-Boltzmann distribution, then the symmetry
condition is satisfied and will continue to be satisfied if the new momenta are projected
onto a surface of constraint or modified to give zero total linear orangular momentum
in the system. As a rule of thumb, the displacement ∆ should be chosen to give roughly
40% acceptance probability (Dellagoet al., 2002).
The basic steps of the shooting algorithm can be summarized as follows:

1. Choose an indexjrandomly on the old trajectory Y(T).


2. Generate a random phase space displacement ∆ in order to generate the new
   shooting point xj∆tfrom the old point yj∆t.


3. Integrate the equations of motion backwards in time from the shooting point to
   the initial condition x 0.


4. If the initial condition x 0 is not in the phase space regionA, reject the trial move.


5. If x 0 ∈A, accept the move with probability min[1,f(x 0 )/f(y 0 )]. Note that if
   the distribution of initial conditions is microcanonical rather than canonical (or
   isothermal-isobaric), this step can be skipped.


6. Integrate the equations of motion forward in time to generate the final point xn∆t.


7. If xn∆t∈B, accept the trial move, and reject it otherwise.


8. If the path is rejected at steps 4, 5, or 7, then the old trajectory Y(T) is counted
   again in the calculation of averages over the transition path ensemble. Otherwise,
   invert the momenta along the backward path of the path to yield a forward moving
   transition path X(T) and replace the old trajectory Y(T) by the new trajectory
   X(T).
   Another important point to note about the transition path samplingapproach is
   that an initial transition path X 0 (T) is needed in order to seed the algorithm. Gen-
   erating such a path can be difficult, particularly for extremely rare event processes.
   However, a few tricks can be employed, for example running a system at high tem-
   perature to accelerate a process or possibly starting a path inBand letting it evolve
   toA. The latter could be employed, for example, in protein folding, whereit is gen-
   erally easier to induce a protein to unfold than fold. Although initial paths generated
   via such tricks are not likely to have a large weight in the transition path ensemble,
   they should quickly relax to more probable paths under the shootingalgorithm. How-
   ever, as with any Monte Carlo scheme, this fast relaxation cannot be guaranteed if
   the initial path choice is a particularly poor one. Just as configurational Monte Carlo