Ensemble average of thermodynamic properties

The probability distribution function and the probability density p (X) should be independent of the ensemble size Af in order to be properly debned. 

Ensemble theory is the crowning achievement of Josiah Willard Gibbs. With ensemble theory he was able to provide for the first time a lucid connection between microscopic states and macroscopic observables. In this section we describe how. Consider any observable, thermodynamic or mechanical property M of a system with N particles. We have argued that any such property is a function of positions and momenta of the particles in the system. 

We define the ensemble average of this property as the average over a large collection Af of macroscopically identical systems  

For arbitrarily large Af we can define a continuous distribution function D(X) and a continuous probability density p(X). 

Let us revisit the previous simple example and consider M = M(X) to be a function of microscopic states. For example, let us assume that M(X) = X. The probability density is a discrete function and the ensemble average of M is given by 

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Ensemble average of thermodynamic properties The probability density is... 

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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