# Canonical averages in a Fermi gas aka generalized Fermi-Dirac distribution

I am in the process of applying Beenakker's tunneling master equation theory of quantum dots (with some generalizations) to some problems of non-adiabatic charge pumping. As a part of this work I encounter thermal averages of single-particle quantities with a fixed total number of electrons. They are pretty straightforward to derive, as recently discussed on Physics.SE, see Combinatorial sum in a problem with a Fermi gas.

I have trouble finding references to prior work for this basic quantum statistics problem. Can you suggest some relevant references in this context?

Progress report:

Writing this stuff up, I realized that I'm basically asking for finite electron number and finite level spacing generalization of the Fermi function:

$\langle \nu_k \rangle = Z_n^{-1} \sum_{n=0}^{n-1} (-1)^{n-m} Z_m e^{-\beta \epsilon_k( n-m)}$ where $Z_n$ is the $n$-electron canonical partition function.

For either $n \gg 1$ or and $\epsilon_{k+1}-\epsilon_k \ll \beta^{-1}$ this reduces to the standard Fermi-Dirac distribution:

$\langle \nu_k \rangle= \frac{1}{1+z_0 e^{-\beta \epsilon_k}}$

The advantage of the new formula is that it only depends on $k$ via $e^{-\beta \epsilon_k}$ with all the combinatorics hidden in $Z_n$. This is unlike the textbook expression summarized in Wikipedia, or Beenakker's implicit function $F(E_p |n)$ (which is $\langle \nu_p \rangle$ in notation).

The question still stands: I don't believe the first formula above is new, what is the right reference?

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