Sampling the Transition Path Ensembles: Trial Moves

So, we see now that, using the factorization trick, we have two ensembles to average over:

$$f_{AB}\left(x_0,t\right) : \mbox{\begin{minipage}{8cm} distn. fcn. of all paths starting at... ... in $B$ at time $t$, used to compute $P\left(\lambda,t\right)$ \end{minipage}}$$ (302)

$$F_{AB}\left(x_0,T\right) : \mbox{\begin{minipage}{8cm} distn. fcn. of all paths starting at... ...\dot{h}_B(t)\right>_{AB}$ and $\left<h_B(t^\prime)\right>_{AB}$.\end{minipage}}$$ (303)

We can compute averages in these two ensembles by MC sampling. A new path is generated from an old path, and is accepted or rejected based on a detailed balance criterion. Now, we sample these two path ensembles in two different simulations, but the acceptance rules and trial moves are the same. Let's generall call $\mathscr{N}\left(i\right)$ the probability of path $i$. We start with an old path ``$o$'' and attempt to generate a new path ``$n$''. The acceptance rule obeying detailed balance is

$$\frac{{\rm acc}\left(o\rightarrow n\right)}{{\rm acc}\left(n... ...)}{\mathscr{N}\left(o\right)\alpha\left(o\rightarrow n\right)}$$ (304)

$\alpha\left(o\rightarrow n\right)$ is the a priori probabilit of attemptying to generate $n$. The moves used in transition path sampling MC guarantee $\alpha\left(o\rightarrow n\right) = \alpha\left(n\rightarrow o\right)$. So,

$$\frac{{\rm acc}\left(o\rightarrow n\right)}{{\rm acc}\left(n... ... = \frac{\mathscr{N}\left(n\right)}{\mathscr{N}\left(o\right)}$$ (305)

So, what are the moves? There are two basic moves we can consider here.

Shooting

Given path $o$, pick a configuration intermediate between $A$ and $B$, rotate all momenta by a little angle randomly selected from [-$\Delta\phi$,$\Delta\phi$], and then MD integrate forwards and backwards to generate a new path. We accept if the backwards integration lands in A and the forwards in B. We get a high acceptance rate because $A$ and $B$ ``attract'' trajectories.

Shifting

Given path $o$, chop off the first few $\Delta t$'s worth of its trajectory, and MD integrate from the end of another $\Delta t$. If the beginning and end are still in $A$ and $B$ respectively, we accept the move. Note that this does not sample ergodically, but it does improve statistics over shooting alone. $\Delta t$ can be positive or negative.

It seems the real trick is generating an initial path of length $T$ that successfully connects region $A$ with region $B$. This can be done with traditional MD by beginning a trajectory in state $A$ and simply waiting long enough for it to cross into $B$. This might be possible, but could easily be prohibitively expensive. A better technique is to guess a path by creating a configuration which you hypothesize to be a transition state, and then integrating forward and backward in time to generate a path. It is accepted as an initial path if the beginning lands in $A$ and the end in $B$.