Bose-Einstein statistics describe how bosons, which do not obey the Pauli exclusion principle, distribute themselves across the available states in a system at equilibrium.
Consider a single-particle state , which can contain any number of bosons. Since the occupation number is variable, we use the grand canonical ensemble, whose grand partition function is as shown below, where is the energy per particle, and is the chemical potential. We evaluate the sum in as a geometric series:
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 Bose-Einstein distribution :
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.