Bose–Einstein statistics

The Bose -Einstein statistics or Bose - Einstein distribution, named after Satyendranath Bose and Albert Einstein, is a probability distribution in quantum statistics ( there also the derivation ). It describes the mean occupation number of a quantum state of energy in thermodynamic equilibrium at absolute temperature for identical bosons as filled particles.

Analog exists the Fermi -Dirac statistics, which passes as well as the Bose -Einstein statistics in the limit of large energy in the Boltzmann statistics for fermions.

Crux of the Bose -Einstein statistics is that while confusing of all four variables of two bosons ( and: local variable;: spin variable), the wave function or state vector of a many-body system does not change sign, while in the Fermi -Dirac statistics very well changes. In contrast to fermions can therefore be several bosons in the same one-particle state, ie have the same quantum numbers.

In interaction freedom

In interaction freedom ( Bose gas ) results for bosons the following formula:


  • The chemical potential, which is always less than the lowest possible energy value for bosons :; Therefore, the Bose -Einstein Statistk is only defined for energy values ​​.
  • The energy normalization. The choice of temperature depends on the scale used: usually it is chosen to be the Boltzmann constant;
  • It is, when the temperature in units of energy, such as joule, is measured; this happens when in the definition of entropy - does not show up - which is then unitless.

Below a very low critical temperature is obtained at interaction freedom - under the assumption that tends to the energy minimum - the Bose -Einstein condensation.

Note that it is is the occupation number of a quantum state. If one needs the occupation number of a degenerate energy levels, so the above expression is additionally multiplied by the appropriate degree of degeneracy (: spin, bosons with always an integer ), see also multiplicity.

Pictures of Bose–Einstein statistics