Some Theoretical Background to the Metropolis Method

The Metropolis algorithm generates a Markov chain of states. A Markov chain satisfies the following two conditions  

The outcome of each trial depends only upon the preceding trial and not upon any previous trials. 

Each trial belongs to a finite set of possible outcomes. 

Condition (1) provides a clear distinction between the molecular dynamics and Monte Carlo methods, for in a molecular dynamics simulation aU of the states are connected in time. Suppose the system is in a state m. We denote the probability of moving to state n as 7t , . The various 7r , can be considered to constitute an N x N matrix Jt (the transition matrix), where N is the number of possible states. Each row of the transition matrix sums to 1 (i.e. the sum of the probabilities 7r , for a given m equals 1). The probability that the system is in a particular state is represented by a probability vector p  

Thus Pi is the probability that the system is in state 1 and p the probability that the system is in state m. If p(l) represents the initial (randomly chosen) configuration, then the probability of the second state is given by  

---