The Gibbs-Boltzmann Probability Density

Observe that the previous discussion does not quite tell us the form of the phase space density associated to the canonical ensemble. An elegant way to derive this 

This can be seen as a variational principle. Using the method of Lagrange multipliers, we seek stationary solutions of 

The normalization condition determines the parameter A, whereas and hence the temperature arises from the choice of average energy, i.e. from solving the equation 

Alternatively, we may take the position that the temperature is specified and the mean energy is a direct consequence of this choice. 

This defines the phase space density of the canonical ensemble which we refer to as the Gibbs-Boltzmann density  

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