General Principles of the Monte Carlo Method

Most of the applications of the Monte Carlo method to polymer systems refer to the canonical ensemble (N,V,T). The canonical probability that a system of N particles in a volume V at a temperature T is found in a configuration X is proportional to its Boltzmann weighted energy at that state E and it is given by 

Step 4 Compare the weight ct to a (uniform) random number r in the 

The result of this algorithm is a Markov chain in which a probability of a configuration is proportional to cr. The sum EFJNs will give the average value of f with N5 being the (usually large) number of simulation steps the summation is over all steps where the function is evaluated. 

There are two important points that need to be highlighted here (1) the sampling method must be ergodic, and (2) the sampling method should not introduce improper bias. One may employ transition probabilities that satisfy the following conditions  

This way the generated configurations are included with probability which is obtained from the transition probabilities The Metropolis algorithm is a subset of the use of Markov chains to sample configuration space. 

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