Probability functionals, transition path

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%... 

The probability PXqr q ) fo move the reaction coordinate centroid variable from the reactant configuration to the transition state is readily calculated [108] by PIMC or PIMD techniques [17-19] combined with umbrella sampling [77,108,123] of the reaction coordinate centroid variable. In the latter computational technique, a number of windows are set up which confine the path centroid variable of the reaction coordinate to different regions. These windows connect in a piecewise fashion the possible centroid positions in going from the reactant state to the transition state. A series of Monte Carlo calculations are then performed, one for each window, and the centroid probability distribution in each window is determined. These individual window distributions are then smoothly joined to calculate the overall probability function in Eq. (4.11). An equivalent approach is to calculate the centroid mean force and integrate it from the reactant well to barrier top (i.e., a reversible work approach for the calculation of the quantum activation free energy [109,124]). 

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. 

The particle moves with a certain probability, which is referred to as transition probability. The transition probability can be a function of the position, i.e., of a box, or can be a function of the time, or both. In general, the transition probability can be a function of the whole history of what happened to the particle. So the transition probability can be dependent on the whole path that the particle went before. 

An alternative definition of the transition state (not based on the commitment probability) was recently introduced by Hummer who quantifies the difference between equilibrium distribution functions and the distribution functions obtained over transition paths (TP). Two regions corresponding to the reactants A and products B are defined in the entire configuration space. A path in this space is assumed to be reactive (transition path) if it starts from A and reaches B without returning to A. A conditional probability P TP x) of being on a transition path, given that the system is at point x of the configuration space, is introduced. This probability is related to the equilibrium distribution Peq x) and the conditional distribution P x TP) of x over transition paths by Eq. [19] ... 

The evolution of a p-automaton is the following if the system is in the state x, the transition e to the state Xj is done with the probability P(x,.,e,x ). The probability transition function can be extended to the paths it c X(ZX) in the p-automaton Ap (a path is obtaining by concatenating the transitions, where the end state and the initial state of two consecutive transitions coincide). The probability of the paths is defined by the next equation ... 

By employing path variables, a Path Probability Function, P, which corresponds to a free energy of the CVM, is written as the product of three teims, Pi, P2 and P3. Each term for disorder-Llo transition is given in the following logarithmic expression. 

In this case, the transition path probability is simply given by a product of delta functions. 

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

In Figs. 4.1 and 4.2, the broken lines do not represent the sample paths of the process X(t), but join the outcoming states of the system observed at a discrete set of times f, t2,.. . , tn. To understand the behavior of X(t), it is necessary to know the transition probability. In Fig. 4.3 are given numerical simulations of a Wiener process W(t) (Brownian motion) and a Cauchy process C(t), both supposed one dimensional, stationary, and homogeneous. Their transitions functions are defined... 

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