Selection and acceptance probabilities

In order to correctly satisfy the detailed balance condition (4.77) in a Monte Carlo simulation, we have to take into account that each Monte Carlo step consists of two parts. First, a Monte Carlo update of the current state is suggested and second, it has to be decided whether or not to accept it according to the chosen sampling strategy. In fact, both steps are independent of each other in the sense that each possible update can be combined with any sampling method. Therefore, it is useful to factorize the transition probability t(X X ) into the selection probability 5(X X ) for a desired update from X to X and the acceptance probability a(X X ) for this update  

The expression (4.80) for the acceptance probability naturally fulfills the detailed-balance condition (4.77). The selection ratio a(X, X ) is unity, if the forward and backward selection probabilities are identical. This is typically the case for simple local Monte Carlo updates. If, for example, the update is a translation of a Cartesian coordinate, x =x + Ax, where Ax e [—xq, - -xo] is chosen from a uniform random distribution, the forward selection for a translation by Ax is equally probable to the backward move, i.e., to translate the 

Monte Carlo and chain growth methods for molecular simulations 

Note that the overall efficiency of a Monte Carlo simulation depends on both a model-specific choice of a suitable set of moves and an efficient microstate sampling strategy based on w(X X )- 

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