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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:
Photon statistics is the theoretical and experimental ... condition for super-Poisson statistics is to use Mandel's formula. [3] ... the Bose-Einstein ...
This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics. The critical temperature depends on the density. A more concise and experimentally relevant [ 19 ] condition involves the phase-space density D = n λ T 3 {\displaystyle {\mathcal {D}}=n\lambda _{T}^{3}} , where
Bosons are quantum mechanical particles that follow Bose–Einstein statistics, or equivalently, that possess integer spin.These particles can be classified as elementary: these are the Higgs boson, the photon, the gluon, the W/Z and the hypothetical graviton; or composite like the atom of hydrogen, the atom of 16 O, the nucleus of deuterium, mesons etc. Additionally, some quasiparticles in ...
Similarly the Bose–Einstein correlations between two neutral pions are somewhat stronger than those between two identically charged ones: in other words two neutral pions are “more identical” than two negative (positive) pions. The surprising nature of these special Bose–Einstein correlations effects made headlines in the literature. [5]
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 ...
The grand canonical ensemble provides a natural setting for an exact derivation of the Fermi–Dirac statistics or Bose–Einstein statistics for a system of non-interacting quantum particles (see examples below). Note on formulation
This formula, apart from the first vacuum energy term, is a special case of the general formula for particles obeying Bose–Einstein statistics. Since there is no restriction on the total number of photons, the chemical potential is zero.