The Generalized Metropolis Monte Carlo Algorithm

In the gMMC algorithm successive configurations of the system are generated to build up a special kind of random walk called a Markov 

We define K q- qf) to be the conditional probability that a configuration at qi will be brought to qj in the next step of the random walk. This conditional probability is sometimes called the transition rate. The probability of moving from q to cf (where q and q are arbitrarily chosen configurations somewhere in the available domain) is therefore given by P( —  

Satisfying the detailed balance condition ensures that the configurations generated by the gMMC algorithm will asymptotically be distributed according to p( ). 

The transition rate may be written as a product of a trial probability II and an acceptance probability A 

From the detailed balance condition, it is straightforward to show that the ratio of acceptance probabilities, given by r, is 

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