  We present an analysis of the use of the Darwin-Fowler approximation in
connection with the statistical IPM, by comparing the results of our recent
studies with the occupation number approach (OCN) and some traditional
statistical independent particle model (IPM) approaches. The analysis of level
density works based on the statistical IPM reveals that the use of the the
Darwin-Fowler approximation, in some of them, is theoretically inconsistent and
some of their results should rather be considered as theoretical coincidences
with other consistent approaches, than proofs of their validity. We conclude
that, in general, the use of the Darwin-Fowler approximation with the
statistical IPM should be used criteriously or, if possible, avoided altogether
and suggest that the combinatorial IPM approaches have important advantages
over the other models and formalisms analyzed in this paper, especially
regarding the consistency of the microscopic description of the nuclear
structure and dynamics of non highly excited systems.
