Monte Carlo Sampling in Path Space

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 

Pgeniz ( ) — z(-I1 (i/ )]. For simplicity, this so-called trial move is often carried out such that the generation probability is symmetric, i.e., such that the probability to generate z nU.-9 ) from z , )( j equals the probability to generate z ( ) from z-n sr) 

In the second stage of each Monte Carlo step the new (or trial) pathway is accepted with a certain acceptance probability Pacc H -) The total proba- 

The detailed balance condition (7.12) can now be satisfied by selecting an appropriate acceptance probability. By inserting the product in (7.14) into the detailed balance condition we find that for a symmetric generation probability the acceptance probabilities for the forward and the reverse move must be related by 

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