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Both Fermi–Dirac and Bose–Einstein become Maxwell–Boltzmann statistics at high temperature or at low concentration. Bose–Einstein statistics was introduced for photons in 1924 by Bose and generalized to atoms by Einstein in 1924–25. The expected number of particles in an energy state i for Bose–Einstein statistics is:
Fermi–Dirac statistics is most commonly applied to electrons, a type of fermion with spin 1/2. A counterpart to Fermi–Dirac statistics is Bose–Einstein statistics, which applies to identical and indistinguishable particles with integer spin (0, 1, 2, etc.) called bosons.
An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas (Fermi–Dirac statistics for fermions, Bose–Einstein statistics for bosons), however it is much more generally applicable than that. The grand canonical ensemble may also be used to describe classical ...
As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on Fermi–Dirac statistics and Bose–Einstein statistics, these principles are extended to large number of particles, with qualitatively similar results.
This is the first quantization approach and historically Bose–Einstein and Fermi–Dirac correlations were derived through this wave function formalism. In high-energy physics , however, one is faced with processes where particles are produced and absorbed and this demands a more general field theoretical approach called second quantization .
All known particles obey either Fermi–Dirac statistics or Bose–Einstein statistics. A particle's intrinsic spin always predicts the statistics of a collection of such particles and conversely: [3] integral-spin particles are bosons with Bose–Einstein statistics, half-integral-spin particles are fermions with Fermi–Dirac statistics.
In each case the value = gives the thermodynamic average number of particles on the orbital: the Fermi–Dirac distribution for fermions, and the Bose–Einstein distribution for bosons. Considering again the entire system, the total grand potential is found by adding up the Ω i for all orbitals.
Quantum particles are either bosons (following Bose–Einstein statistics) or fermions (subject to the Pauli exclusion principle, following instead Fermi–Dirac statistics). Both of these quantum statistics approach the Maxwell–Boltzmann statistics in the limit of high temperature and low particle density.