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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.
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.
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.
Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics we use the Thomas–Fermi approximation (gas in a box) and go to the limit of a very large trap, and express the degeneracy of the energy states as a differential, and summations over states as integrals.
Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics, and considering the limit of a very large box, the Thomas–Fermi approximation (named after Enrico Fermi and Llewellyn Thomas) is used to express the degeneracy of the energy states as a differential, and summations over states ...
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 .
In more than two dimensions, the spin–statistics theorem states that any multiparticle state of indistinguishable particles has to obey either Bose–Einstein or Fermi–Dirac statistics. For any d > 2, the Lie groups SO( d ,1) (which generalizes the Lorentz group ) and Poincaré( d ,1) have Z 2 as their first homotopy group .