Fermi-Dirac statistics describe how identical fermions, which obey the Pauli exclusion principle, distribute themselves across the available states in a system at equilibrium.
Consider one single-particle state , which can contain or fermions. Because the occupation number is variable, we turn to the grand canonical ensemble, whose grand partition function is as follows, where is the energy of and is the chemical potential:
The corresponding thermodynamic potential is the Landau potential , given by:
The average number of particles in is then found by taking a derivative of :
By multiplying both the numerator and the denominator by , we arrive at the standard form of the Fermi-Dirac distribution or Fermi function :
This gives the expected occupation number of state with energy , given a temperature and chemical potential .
- H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.