Trajectory space, transition path ensemble

The perspective exploited by transition path sampling, namely, a statistical description of pathways with endpoints located in certain phase-space regions, was hrst introduced by Pratt [27], who described stochastic pathways as chains of states, linked by appropriate transition probabilities. Others have explored similar ideas and have constructed ensembles of pathways using ad hoc probability functionals [28-35]. Pathways found by these methods are reactive, but they are not consistent with the true dynamics of the system, so that their utility for studying transition dynamics is limited. Trajectories in the transition path ensemble from Eq. (1.2), on the other hand, are true dynamical trajectories, free of any bias by unphysical forces or constraints. Indeed, transition path sampling selects reactive trajectories from the set of all trajectories produced by the system s intrinsic dynamics, rather than generating them according to an artificial bias. This important feature of the method allows the calculation of dynamical properties such as rate constants. 

Transition path sampling is an importance sampling of trajectories, akin to the importance sampling of configurations described in Section II.E. Specifically, it is a biased random walk in the space of trajectories, in which each pathway is visited in proportion to its weight in the transition path ensemble. Because trajectories that do not exhibit the transition of interest have zero weight in this ensemble, they are never visited. In this way, attention is focused entirely on the rare but important trajectories, those that are reactive. 

We accomplish the random walk through trajectory space as follows Beginning with a trajectory x ° 2T) [here, the superscript (o) stands for old] whose weight in th transition path ensemble is nonzero, we... 

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