Detailed balance condition and acceptance probability

In a Markov-chain Monte Ccirlo scheme, we need to ensure that different points in configuration space cire visited with a frequency proportional to their Boltzmann weight. This is usually achieved by imposing the detciiled bcdance condition 

In the RG algorithm, the transition probability includes the construction of a particulcir tree of trial segments for both the new cind the original state, the stochastic choice of the subset of open segments in both trees, the selection of a particular configuration among the branches  

Recoil growth algorithm for chain molecules with continuous interactions 

Detailed balance is satisfied by imposing the stronger condition of super-detailed balance [29], which means that we obey detailed balance for all possible sets of trees cind open/closed directions of both the new and old configuration ([to, tn, Oo, On]). Many factors in Pg (On I tn.rwti) that are found in the transition probability from the new (n) to the old (o) state cancel with the factors of Pg (O j tn) which are in the expression for the trcinsition probability from the old (o) to the new (n) state. The only remaining term is the probability of generating open directions along the backbone of the tree. Therefore, the correct acceptcince/rejectlon rule is given by 

In this equation, f is the number of trial positions for the first segment, which is equal to 1 when the first segment of the chain is placed at a reindom position in the system. Note that the term f X k is present in both the numerator cind denominator of equation 3.9 and is therefore 

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