In the Monte Carlo method one calculates the expectation value for an observable A by computing the average with respect to an appropriate distribution. The distribution function is determined by the statistical mechanical ensemble. Assume that the system is described by a Hamiltonian H(x) with x being the degrees of freedom of the system. For the canonical ensemble with fixed temperature T, volume V and number of particles N the expectation value for an observable A is given by

where

Here W denotes the phase space, i.e., all configurations that are available to the system. If the number of configurations of the system under the given constraints is very large, the task of evaluating of the above equation becomes formidable and one has to resort to sampling. Sampling here means that we want to pick up mainly those contributions to the integral that make the largest impact. If we were to randomly sample the available phase we would, for the most part, obtain states that give a very small contribution to the expectation value. We cannot apply simple sampling to a distribution of states that is sharply peaked. To sample the major contributions of the integrant to the integral one constructs a Markov chain of states where each state or configuration is generated from the previously generated configuration:

with . The state is derived from the state . This is not done deterministically as for the integration of motion of the equations of motion in newtonian dynamics, but probabilistically. The state follows from the state with a probability. There is a transition probability

from one state to the other. If the system is in state , and it is so with a probability , then the state changes to the state with the probability W. This evolution seems to be quite different from what we have learned in the preceding chapter. There, the evolution was deterministic. Given the initial conditions the entire evolution of the states of the system is determined for ever.