Sampling of probability distribution functions

The canonical ensemble average of any thermodynamic property of matter, M(p, q), is 

For example, the excess internal energy for non-ideal systems is the ensemble average of the potential energy 

Remembering that the integral over the phase space is an approximation of the sum over phase space points, we can write 

For dense systems it is not possible to determine the integrals in Eq. 15.2, or the sums in Eq. 15.3 because the total number of the configurational phase space points representing nn NVT system at the thermodynamic limit is stupendously large. Thankfully, as in any other statistical average, it is sufficient to sample the phase space and estimate the ensemble average of thermodynamic properties. If the phase space is sampled sufficiently well, the ratio of the sums in Eq. 15.3 can converge 

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