Fermi gas at the absolute zero T

[JH_Park]

I'm currently studying Thermodynamic properties of a Fermi gas at the absolute zero temperature.
I get how the internal energy, pressure... etc of the gas are derived. For example, in computing the internal energy, one sums up all the energy of states weighted by its average occupation number(which is derived from using grand-canonical ensemble formalism) of that states, including a degeneracy of the states.
U=∑(2S+1)ε(k)n(k)
(where S: total spin, ε(k): energy of a standing fermi wave whose wave vector is k,
n(k): average occupation number of k)
I shall call this method as "method 1". This approach intuitively makes sense to me but I found a little bit of subtle mathematical inconsistency in this calculation.
As far as I understand, when we compute an average of some observable quantity in Statistical Mechanics, we first calculate an appropriate partition function depending on which ensembles we use. Then, from the partition function, we next compute the corresponding thermodynamic potentials and by making appropriate derivatives we can get average values of thermodynamic quantities of the ensemble. (method 2)
For consistency, I personally want to derive thermodynamic quantities using the method 2 but all the textbook I reference at follow the method 1. Is there any mathematical proof I can look up saying that method 1 and 2 are essentially identical?? or Is there any reference that actually derives thermodynamic quantities using the method 2?
Sorry for bad english TT

 

[DrClaude]

The problem of method 2 is that you have to ensure that the sum over states only includes states that are allowed by the Pauli exclusion principle. Starting from the grand canonical ensemble to derive a canonical ensemble solution allows to circumvent that difficulty, by considering only one single-particle state at a time.