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Comparison of the Boltzmann distribution with the Fermi distribution and Bose distribution. All three distributions give occupation probability \(P(W)\) as a function of the energy \(W\) of the gas particle. For large energies \(W\) of the gas particles under consideration, which can be achieved by a high temperature \(T\), the distributions do not differ. Thus, in the high temperature limit, the classical Boltzmann distribution can be used to describe the Fermi gas (for example an electron gas) or the Bose gas (for example a photon gas).
On the other hand, at lower energies \(W\), that is at low temperatures, quantum gases differ from the classical gas described by the Boltzmann distribution. In this case, the Fermi distribution for fermions or Bose distribution for bosons must be used.