Generation probability, transition path

This detailed balance condition makes sure that the path ensemble sg[z )] is stationary under the action of the Monte Carlo procedure and that therefore the correct path distribution is sampled [23, 25]. The specific form of the transition matrix tt[z(° 2 ) -> z(n, 9-) depends on how the Monte Carlo procedure is carried out. In general, each Monte Carlo step consists of two stages in the first stage a new path is generated from an old one with a certain generation probability... 

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. 

In this equation, Pg (i —> j) is the probability to generate a trial move from state i to state j. When one assumes that the total energy along the transition path is conserved exactly, one is able to compute the second factor on the r.h.s. on the last line of equation 6.17 without having to integrate the equations of motion. This means that one can reject paths with an unfavorable energy immediately [236]. 

Select a random point (here isobutcine molecule) of a transition path. For this point [x, Ukinl. we generate ui random momenta p from a Maxwell-Boltzmcinn distribution. These momenta are rescaled in such a way that the toted initial kinetic energy of the system (Ukin) is a constant. The value of ni should be large enough to have some reasonable initial statistics on the probability of reaching regions A and B. 

By concatenating Monte Carlo transition probabilities according to Eq. (1.1), one obtains the probability of a particular stochastic path x. ) generated in a Metropolis Monte Carlo simulation. The time variable t describing the progress of this stochastic process is artificial. This Monte Carlo time can be approximately mapped to a physical timescale by comparing known dynamical properties such as transport coefficients [8,41]. 

A local algorithm for Metropolis Monte Carlo trajectories must be constructed carefully. Due to the finite probability of exactly repeated states in these paths, the corresponding transition probability includes a singular term [see Eq. (1.14)]. The generation algorithm for local path moves must take this singularity into account properly. Appropriate acceptance probabilities are given in [5]. (H. C. Andersen has drawn our attention to an omission in [5]. In Metropolis Monte Carlo trajectories sequences of multiple rejections can occur. Attempts to modify time slices in the interior... 

The convergence property of the nonadiabatic transition amplitudes with respect to the number of branching generation is next studied. We here show the results only for the three-state model, since the two state model offers only an easier test. The transition probability of PSANB has been calculated with Eq. (6.78) with use of the branching paths of Fig. 6.5(a). This approximation is justified because the endpoints of the paths are generally very close to each other on the individual potential curves. To estimate the accuracy, we compare the probabilities with those obtained in the full quantum calculations described above. 

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