Bose-Einstein Distribution Function
An orbital can support any number of bosons, which fundamentally changes
the Gibbs Sum and thus the distribution function. Instead of summing
over N = 0, 1 we must sum over all N. The final result is:
f (
) =
Einstein Condensation
Since there is no restriction on the number of particles in the ground
state, a low enough temperature would deny the system of the thermal
excitation required to promote very many bosons out of the lowest energy
orbital.
There is, then, a transition temperature below which the lowest energy
"ground" orbital possesses a large number of bosons. Above this
temperature, entropy and thermal excitation render the ground orbital
sparsely populated. This transition temperature is known as the
Einstein condensation temperature, and the effect of bosons crowding
the ground orbital is known as the Einstein condensation.
The Einstein condensation temperature is given by:
τ_{}âÉá
(
)
^{2/3}
The most common condensate is liquid Helium. The crowding is so
profound that one can actually see macroscopically the ground orbital of
a Helium liquid with the proper equipment. Physics such as
superfluidity are also outgrowths of the study of this condensation.