a mixture of two gases, one made out of free electrons and the other one consisting of PP. In effect, on account of (\[Z equals Z0 times dZ\]) the total electron density $ n\left( \beta ,\mu \right) =L^{-3}k_{B}T\partial \ln Z/\partial \mu  $ can be written as$$\label{gaussian-density-eq}
n\left( \beta ,\mu \right) =n_{0}\left( \beta ,\mu \right) +\delta n\left( \beta ,\mu \right)$$ where the first and second terms on the right hand side come from $ Z_{0} $ and $ \delta Z $, respectively. In particular the contribution from free electrons has the familiar form $ n_{0}\left( \beta ,\mu \right) =2L^{-3}\sum _{\mathbf{k}}f\left( \beta \varepsilon _{\mathbf{k}}\right)  $ where $ f\left( x\right) \equiv \left( e^{x}+1\right) ^{-1} $ is the Fermi distribution function. As is well known in the BCS limit $ \mu \beta \rightarrow \infty  $ all electrons are unpaired just above $ T_{c} $ so we have $ n\approx n_{0} $. Conversely, in a dilute system with strong attraction (BE limit) we have $ n\approx \delta n $: the normal state just above $ T_{c} $ is composed exclusively of preformed pairs, whose BE condensation leads to superconductivity.

It is important to note that the above functional-integral theory is only valid either at weak 