The Absolute Entropy and Free Energy as Ensemble Averages

The Absolute Entropy and Free Energy as Ensemble Averages 

P (x) enables one to define ensemble averages of various thermodynamic quantities that are defined over the configurational space. For example, the average configurational energy is 

Note that in the terminology of the theory of statistics, a function, such as (x), that is defined over fi, is called a random variable the brackets denote a statistical average defined with the Boltzmann PD. We now define a total absolute entropy S that depends on both the coordinates and the momenta (see Eqs. [IJ and [2],  

For simplicity, throughout this chapter we shall always use the common notation S rather than S . It should be pointed out that whereas E (Eq. [71) is a well-defined function, S (Eq. [9J) is defined up to an additive constant that depends on the units used for x. Equation [8] is useful because it enables one to calculate the contribution of the momenta to the entropy. 

The configurational Helmholtz free energy F can also be expressed formally as a statistical average, F , of the random variable [ (x) + k T InPB(x)], 

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