Fermi–Dirac statistics

Comparison of average occupancy of the ground state for three statistics

The Fermi–Dirac distribution approaches the Maxwell–Boltzmann distribution in the limit of high temperature and low particle density, without the need for any ad hoc assumptions:

and the average particle number for that single-particle level substate is given by

For example, distributing two particles in three sublevels will give population numbers of 110, 101, or 011 for a total of three ways which equals 3!/(2!1!).