Transition path ensemble

The basis of the transition path sampling method is the statistical description of dynamical pathways in terms of a probability distribution. To define such a distribution consider a molecular system evolving in time and imagine that we take snapshots of this system at regularly spaced times fj separated by the time step At. Each of these snapshots, or states, consists of a complete description z of the system in terms of the positions q = 71, 72, , In and momenta p = pi,P2, , pn of all N atoms in the system, 2 = q, p. If we follow the system for a total time A we obtain an ordered sequence of L = 3 /At + 1 states 

This sequence of states is a discrete representation of the continuous dynamical trajectory starting from zo at time t = 0 and ending at z at time t = . Such a discrete trajectory may, for instance, result from a molecular dynamics simulation, in which the equations of motion of the system are integrated in small time steps. A trajectory can also be viewed as a high-dimensional object whose description includes time as an additional variable. Accordingly, the discrete states on a trajectory are also called time slices. 

The probability to observe a particular sequence of states depends on the distribution of the initial conditions and the dynamical rule describing the time evolution 

The first factor on the right-hand side of the above equation, p(z0), is the distribution of initial conditions zo, which, in many cases, will just be the equilibrium distribution of the system. For a system at constant volume in contact with a heat bath at temperature T, for instance, the equilibrium distribution is the canonical one 

The specific form of the short-time transition probability depends on the type of dynamics one uses to describe the time evolution of the system. For instance, consider a single, one-dimensional particle with mass m evolving in an external potential energy V(q) according to a Langevin equation in the high-friction limit 

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Just as in a conventional Monte Carlo simulation, correct sampling of the transition path ensemble is enforced by requiring that the algorithm obeys the detailed balance condition. More specifically, the probability n [ZW( ) - z(n)( )]2 to move from an old path z ° 22) to a new one " (2/ ) in a Monte Carlo step must be exactly balanced by the probability of the reverse move from 22) to z<,J> 22)... 

Then the path probability from (7.3) consists of a product of such delta functions. Due to the singular nature of such a path probability it is more convenient to view the entire deterministic trajectory as represented by its initial state z0. In this case the transition path ensemble from (7.10) reduces to a distribution of initial conditions z0 yielding pathways connecting srf with 2%... 

Thus, the transition path ensemble is now represented by a distribution of initial conditions in phase space (we have, in effect, integrated out all path variables except... 

Dellago, C. Bolhuis, P. G. Chandler, D., On the calculation of rate constants in the transition path ensemble, J. Chem. Phys. 1999,110, 6617-6625... 

The transition path ensemble (3) is a complete statistical description of all possible pathways connecting reactants with products. Pathways sampled according to this ensemble are typical trajectories which can then be analyzed to yield information about mechanisms and rates. The definition of the transition path ensemble is very general and valid for all Markovian processes. In the following we will write down the specific form of the transition path ensemble for different types of processes. 

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... 

Accept or reject the new pathway according to a Metropolis acceptance criterion obeying detailed balance with respect to the transition path ensemble... 

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