Bose–Einstein statistics

The Bose-Einstein statistic or also Bose-Einstein distribution, named after Satyendranath Bose (1894-1974) and Albert Einstein (1879-1955), is a probability distribution in quantum statistics (there also the derivation). It describes the mean occupation number ⟨ \langle n(E) \rangle a quantum state of energy E\,in thermodynamic equilibrium at absolute temperature Tfor identical bosons as occupying particles.

Analogously, for fermions there exists the Fermi-Dirac statistics, which, like the Bose-Einstein statistics, merge into the Boltzmann statistics in the limiting case of large energy E

The key point of the Bose-Einstein statistics is that if all four variables x, y, z, m\,two bosons ( x, y\,and z\,: location variable; m\,: spin variable) the wave function ψ \psi \,or the state vector of a many-particle system does not change sign (\psi \rightarrow \psi) , whereas in the Fermi-Dirac statistics it does change (\psi \rightarrow -\psi) . Therefore, unlike fermions, several bosons can be in the same one-particle state, i.e., have the same quantum numbers.

Zoom

Occupation number ⟨ \langle n \rangleas a function of the energy E - \mu
for bosons (Bose-Einstein statistics, upper curve)
or fermions (Fermi-Dirac statistics, lower curve),
in each case in the special case of interaction freedom and at constant temperature
The chemical potential μ \mu is a parameter that depends on temperature and density;
in the Bose case it is always smaller than the energy and would vanish in the limiting case of Bose-Einstein condensation;
in the Fermi case, however, it is positive, at T = 0 \, \mathrm{K}it corresponds to the Fermi energy.

With freedom from interaction

If there is no interaction (bosegas), the following formula results for bosons:

\langle n(E) \rangle = \frac {1}{e^{\beta (E - \mu)} - 1}

with

  • the chemical potential μ \mu which for bosons is always smaller than the lowest possible energy value: μ therefore\mu < E
    the Bose-Einstein statistics is
    only for energy values E - μ
  • of the energy normalization β \beta . The choice of β \beta temperature scale used:
    • usually it is chosen to be β \beta = 1/(k_\mathrm{B} T)with Boltzmann constant k_{\mathrm {B} };
    • it is β \beta = 1/Twhen the temperature is measured in energy units, such as joules; this occurs when k_{\mathrm {B} }also does not appear in the definition of entropy-which is then unitless.

Below a very low critical temperature T_\lambdaone obtains Bose-Einstein condensation under interaction freedom - assuming that μ \mu \,tends towards the energy minimum.

Note that ⟨ \langle n(E) \rangle the occupation number of a quantum state. If one needs the occupation number of a degenerate energy level, the above expression has to be multiplied additionally by the corresponding degeneracy g_i = 2s +1( s\,: spin, for bosons always integer), cf. also multiplicity.


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