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

+ 7 like - 0 dislike
51 views

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?

This post has been migrated from (A51.SE)
retagged Apr 19, 2014

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ys$\varnothing$csOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.