Path ensemble

Crooks, G.E., Path-ensemble averages in systems driven far from equilibrium, Phys. Rev. E 2000, 61, 2361-2366... 

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)... 

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... 

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... 

The notation ( ) a used in the last equality of the above equation indicates that this expression can be viewed as the average work in the so-called work-weighted path ensemble... 

In the work-weighted path ensemble the statistical weight of a particular trajectory z 3 ) explicitly depends on the work performed on the system along that trajectory. The work distribution in the work-weighted path ensemble for a particular value of... 

Since the bias function should enhance the sampling of pathways with important work values it can be made to depend on the work only, ir[z 2 ) = n W( (. Z))]. To minimize the statistical error in the free energy difference the bias function needs to be selected such that both the statistical errors of the numerator and the denominator of (7.44) are small. Ideally, the bias function should have a large overlap with both the unbiased work distribution P(W) and the integrand of (7.36), P (W) exp (—j3W). Just as Sun s work-biased ensemble Pa[z( ), the biased path ensemble )] can... 

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... 

Since we want to focus on reactive pathways we now restrict the path ensemble to those pathways that start in region A, the reactant region, and end in region B, the product region. This is achieved by multiplying the path probability V[x T) with the appropriate characteristic functions ... 

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. 

Fig. 3. In this free energy landscape the coordinate q fails to distinguish the basins of attraction of the stable states. If region A is defined by requiring that q is less than some limit (light gray region), points belonging to the basin of attraction of B are located inside A. Path starting from such points (such as the dark gray path) belong to the path ensemble but are not real reactive trajectories. For a correct definition of the stable states also the variable q must be taken into account...

Various deterministic and stochastic sampling techniques for path ensembles have been proposed [4-6]. Here we consider only Monte Carlo methods. It is important, however, to be aware that while the path ensemble is sampled with a Monte Carlo procedure each single pathway is a fully dynamical trajectory such as one generated by molecular dynamics. 

To guarantee that pathways are visited with a frequency proportional to their weight in the path ensemble VAslxiT)] we require that the forward move from the old path to the new one must be exactly balanced by the reverse move ... 

---