Bose Einstein statistics

Occupation number as a function of energy for bosons ( Bose-Einstein statistics, upper curve ) or fermions (Fermi-Dirac statistics, lower curve), in each case in the special case of freedom from interaction and at constant temperature . The chemical potential is a parameter that depends on temperature and density; in the Bose case it is always smaller than the energy and would disappear in the limit case of the Bose-Einstein condensation ; in the Fermi case, on the other hand, it is positive, in the case it corresponds to the Fermi energy . ${\displaystyle \langle n\rangle }$ ${\displaystyle E-\mu }$


${\displaystyle T>0}$
${\displaystyle \mu }$

${\displaystyle T=0\,\mathrm {K} }$

The Bose-Einstein statistics or 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 of a quantum state of energy  in thermodynamic equilibrium at absolute temperature for identical bosons as occupying particles. ${\displaystyle \langle n(E)\rangle }$ ${\displaystyle E\,}$ ${\displaystyle T}$

Similarly, the Fermi-Dirac statistics exist for fermions , which, like the Bose-Einstein statistics, are transferred to the Boltzmann statistics in the extreme case of high energy . ${\displaystyle E}$

Central point of the Bose-Einstein statistics is that while permutation of all four variables of two bosons ( and : local variable; : Spin variable) the wave function and the state vector of a many-body system not the sign changes , while in the Fermi-Dirac statistics it very probably changes . In contrast to fermions, several bosons can therefore be in the same one-particle state, i.e. have the same quantum numbers . ${\displaystyle x,y,z,m\,}$ ${\displaystyle x,y\,}$ ${\displaystyle z\,}$ ${\displaystyle m\,}$ ${\displaystyle \psi \,}$ ${\displaystyle (\psi \rightarrow \psi )}$ ${\displaystyle (\psi \rightarrow -\psi )}$

If there is no interaction

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

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

With

• the chemical potential  , which is always less for bosons than the lowest possible energy value: ; therefore the Bose-Einstein statistics are only defined for energy values . ${\displaystyle \mu }$ ${\displaystyle \mu <E}$
  ${\displaystyle E-\mu >0}$
• of energy normalization . The choice of depends on the temperature scale used: ${\displaystyle \beta }$ ${\displaystyle \beta }$
  • Usually it is chosen to with the Boltzmann constant ; ${\displaystyle \beta =1/(k_{\mathrm {B} }T)}$ ${\displaystyle k_{\mathrm {B} }}$
  • it is when the temperature is measured in units of energy such as joules ; this happens even if it does not appear in the definition of entropy - which then has no units. ${\displaystyle \beta =1/T}$ ${\displaystyle k_{\mathrm {B} }}$

Below a very low critical temperature , if there is no interaction - assuming that it tends towards the energy minimum - the Bose-Einstein condensation is obtained . ${\displaystyle T_{\lambda }}$ ${\displaystyle \mu \,}$

Note that it is the occupation number of a quantum state. If the occupation number of a degenerate energy level is required, the above expression must also be multiplied by the corresponding degree of degeneracy ( : spin, always an integer for bosons), cf. also multiplicity . ${\displaystyle \langle n(E)\rangle }$ ${\displaystyle g_{i}=2s+1}$ ${\displaystyle s\,}$

literature

• U. Krey, A. Owen, Basic Theoretical Physics - a Concise Overview , Berlin Heidelberg New York, Springer 2007, ISBN 978-3-540-36804-5 (in English)
• LD Landau, EM Lifschitz, Statistical Physics , Verlag Harri Deutsch, former Akademie Verlag Berlin 1987. (uses unusual temperature unit).