# Is there an analogue to Fermi-Dirac statistics for interacting electrons?

Alongside the formula I've seen for Fermi-Dirac statistics: $$\frac{1}{e^{\beta(E-\mu)}+1}=f(E)$$ I often see the addendum that this is only true for non-interacting electrons. By that, I sort of assume it means that two electrons will not interact and occupy the same state (otherwise, if it's simply a question of the forces exerted by the electrons, then this could be surely taken into account with the chemical potential term - or is the issue in the assumption of a constant difference between energy states?)

If we allow the electrons to interact we have a Fermi liquid instead. As you can read in the link, you can consider this a system of free quasiparticles, which are collective excitations of the system (and do not correspond to any actual particles). These quasiparticles will obey Fermi-Dirac statistics, but the actual electrons will have their distribution modified by the "quasiparticle weight" $$Z$$, the exact origin of which requires a more involved discussion.