system
Given the following post, respond as if you're commenting on Reddit:
human
If I am trying to model one single electron (or one hole) delocalized among a small number of molecules, is it appropriate to use a Fermi-Dirac distribution or a Boltzmann distribution for temperature dependence, and why?

Please let me know if any clarification is needed to answer my question.
gpt
Recall that the Fermi-Dirac distribution is the grand canonical ensemble, and the Boltzmann distribution is the canonical ensemble (your terminology is correct, the other answers are confusing "Boltzmann distribution" with "Maxwell-Boltzmann distribution"). So, if your system cannot exchange electrons from elsewhere (either your heat bath or the molecules), and your delocalized energy levels are all unoccupied (and therefore accessible to the electron), then you should use the density matrix diag(e^(-E_n/kT)/Z) in the energy basis. However, if you're trying to look at one electron in the system which also happens to contain many other electrons, you need to take statistics into account with the degeneracy of each level, and it's far easier in Fermi-Dirac (number fluctuations *should* be small for enough particles).

For the other answerers: the assumption that the system is exchanging heat with the environment means that this **is** a system + heat bath of a large number of particles. OP is just focusing on the subsystem (i.e. we are not in the microcanonical ensemble). This is totally standard when modeling a non-interacting state - the multi-particle partition function simply decomposes into its one-particle partition functions.