Ensemble of trajectories

One way to overcome this problem is to start by setting up the ensemble of trajectories (or wavepacket) at the transition state. If these bajectories are then run back in time into the reactants region, they can be used to set up the distribution of initial conditions that reach the barrier. These can then be run forward to completion, that is, into the products, and by using transition state theory a reaction rate obtained [145]. These ideas have also been recently extended to non-adiabatic systems [146]. 

Application of a Stochastic Path Integral Approach to the Computations of an Optimal Path and Ensembles of Trajectories ... 

A related algorithm can be written also for the Brownian trajectory [10]. However, the essential difference between an algorithm for a Brownian trajectory and equation (4) is that the Brownian algorithm is not deterministic. Due to the existence of the random force, we cannot be satisfied with a single trajectory, even with pre-specified coordinates (and velocities, if relevant). It is necessary to generate an ensemble of trajectories (sampled with different values of the random force) to obtain a complete picture. Instead of working with an ensemble of trajectories we prefer to work with the conditional probability. I.e., we ask what is the probability that a trajectory being at... 

If the above assumption is reasonable, then the modeling of most probable trajectories and of ensembles of trajectories is possible. We further discussed the calculations of the state conditional probability and the connection of the conditional probability to rate constants and phenomenological models. 

An ensemble of trajectory calculations is rigorously the most correct description of how a reaction proceeds. However, the MEP is a much more understandable and useful description of the reaction mechanism. These calculations are expected to continue to be an important description of reaction mechanism in spite of the technical difficulties involved. 

The TS trajectory z (f) given by Eq. (21) depends on time t and on the instance of the noise that is represented by the subscript a. Properly speaking, therefore, it defines a statistical ensemble of trajectories. For a fixed time to, it specifies a random variable z (f0) with certain statistical properties. Its... 

The constant matrices i- act as projection operators onto the different eigenspaces. They are given in Ref. 38. The solution Eq. (30) is entirely analogous to Eq. (20) in the white noise case. To obtain a trajectory that remains in the vicinity of the barrier for all times, we again have to set caj = 0 and identify Eq. (31) as the TS trajectory. It satisfies the condition of the general definition in that it provides, at fixed time, a random ensemble of trajectories that is stationary in time, and at fixed noise sequence a a trajectory that spends most of its time close to the barrier. 

This identity [3, 15] between a weighted average of nonequilibrium trajectories (r.h.s.) and the equilibrium Boltzmann distribution (l.h.s.) is implicit in the work of Jarzynski [2], and is given explicitly by Crooks [16]. The average (... is over an ensemble of trajectories starting from the equilibrium distribution at / 0 and... 

This expression may be interpreted as a ratio of two partition functions. In the denominator we have the partition function Z / of all trajectories starting in region with endpoint anywhere the integral in the numerator is the partition function Zai t) of all trajectories starting in. c/ and ending in 38 [this is the normalizing factor of (7.11)]. We can then view the ratio of partition functions as the exponential of the free energy difference between these two ensembles of trajectories... 

Finally, in Sect. 7.6, we have discussed how various free energy calculation methods can be applied to determine free energies of ensembles of pathways rather than ensembles of trajectories. In the transition path sampling framework such path free energies are related to the time correlation function from which rate constants can be extracted. Thus, free energy methods can be used to study the kinetics of rare transitions between stable states such as chemical reactions, phase transitions of condensed materials or biomolecular isomerizations. 

In spite of these potential concerns, the MEHMC method is expected to be a useful tool for many applications. One task for which it might be particularly well suited is to generate a canonical ensemble of representative configurations of a bio-molecular system quickly. Such an ensemble is needed, for example, to represent the initial conditions for the ensemble of trajectories used in fast-growth free energy perturbation methods such as the one suggested by Jarzynski s identity [104] (see also Chap. 5). 

A detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

The answer to the problem lies, I believe, in that a single classical trajectory is never the answer to a dynamical question. There are two reasons. One is a reason of principle. To mimic quantal initial conditions it is necessary to generate an entire ensemble of trajectories, each with somewhat different initial conditions. Then one needs to average over these initial conditions. Now comes a wonderftd technical point (and a tribute to Stan Ulam [27] who, as far as I know, was the first to realize it. The number of initial conditions that one needs to sample is independent of the... 

Furthermore, a related and common criticism of the MFT method is that a mean-field approach cannot correctly describe the branching of wave packets at crossings of electronic states [67, 70, 82]. This is true for a single mean-field trajectory, but is not true for an ensemble of trajectories. In this context it may be stressed that an individual trajectory of an ensemble does not even possess a physical meaning—only the average does. 

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

The electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

Figure 2). The calculations were done in the microcanonical ensemble at a temperature of 300K 5K. Energy was well conserved throughout the trajectories, and no overall drifts in molecular temperature were observed. Small ensembles of trajectories (12 for SI and 6 each for the other minima) were calculated for the averaging of system properties. Each trajectory was equilibrated by velocity reassignments during an initial period of 20ps, followed by another 20ps of dynamics used for data collection.

The master equation evolves the classical degrees of freedom on single adiabatic surfaces with instantaneous hops between them. Each single (fictitious) trajectory represents an ensemble of trajectories corresponding to different environment initial conditions. This choice of different environment coordinates for a given initial subsystem coordinate will result in different trajectories on the mean surface the average over this collection of classical evolution segments results in decoherence. Consequently, this master equation in full phase space provides a description in terms of fictitious trajectories, each of which accounts for decoherence. When the approximations that lead to the master equation are valid, this provides a useful simulation tool since no oscillatory phase factors appear in the trajectory evolution. 

It is a top-down approach in providing explicit recipes for all these geometrical stmctures, and it is presumably equivalent to the bottom-up approach of working through their manifestations in terms of ensembles of trajectories [5]. 

Figure 1. A scheme of separatrices. Ensemble of trajectories are schematized by gray sets. The three sets of trajectories, 1,2,3, evolute but never cross the separatrices represented by dashed lines.

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