Fermi-Dirac statistics

Fermi-Dirac statistics are used to describe a gas of indistinguishable fermions. Particles with half-integral spin must necessarily have wave functions which are anti-symmetric under particle exchange. That is, two configurations which differ by only exchanging the positions and velocities of a pair of fermions are given identical weights times -1. This is a realization of the Pauli exclusion principle.

The average number of particles in state s is given by

<ns> = 1/(exp((Es-u)/ kBT) + 1)

where Es is the energy of a particle in s, u is the chemical potential, T is the temperature, and kB is Boltzmann's constant. Compare this to Bose-Einstein statistics where the +1 is replaced by -1. In the limit where exp((Es-u)/ kBT) >> 1, either due to large T or large u, then the quantum nature of the gas is unimportant and the system is described by classical Maxwell-Boltzmann statistics.