Fermi-Dirac distribution
/ FAIR-mee dih-RAK /
Suppose you have a stack of energy levels and a crowd of electrons to put in them at some temperature. Which seats end up occupied? The Fermi-Dirac distribution is the rulebook that answers this. Because electrons refuse to share a state, they can't all pile into the lowest seats — they fill from the bottom up, and temperature only blurs the top edge.
The rule gives, for each energy, the probability that a seat there is taken. At absolute zero it's a sharp cliff: everything below the Fermi energy is fully occupied (probability one), everything above is empty (probability zero). Warm it up and the cliff softens into a gentle slope a few degrees wide, where some electrons just above the Fermi energy are occupied and some just below are emptied.
It matters because this softening is the source of nearly all temperature effects in metals — heat capacity, conductivity changes, and more. The common misconception is to think heat excites all the electrons; in truth the cliff only blurs over a tiny range, so only the thin band of electrons near the Fermi energy ever feels the temperature at all.
At room temperature in copper, the "blur" at the top of the filled levels is only about a hundredth as wide as the Fermi energy itself. So out of every hundred conduction electrons, only roughly one is in a position to do anything thermal — the rest are locked in place.
At room temperature the occupied-empty edge blurs over only ~1% of the Fermi energy.
At energies far above the Fermi level, the Fermi-Dirac rule fades into the familiar classical curve for a hot gas. So Drude's old picture isn't wrong everywhere — it's just the high-energy tail of the more complete quantum rule.