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Let us now consider BoseEinstein statistics. The particles in the
system are assumed to be massive, so the total number of
particles is a fixed number.
Consider the expression (584). For the case of massive bosons, the numbers
assume all values
for each , subject to
the constraint that .
Performing explicitly the sum over , this expression reduces to

(604) 
where is the partition function for particles
distributed over all quantum states, excluding state , according
to BoseEinstein statistics [cf., Eq. (586)]. Using Eq. (591),
and the approximation (592), the above equation reduces to

(605) 
Note that this expression is identical to (598), except that
is replaced by
. Hence,
an analogous calculation to that outlined in the previous subsection yields

(606) 
This is called the BoseEinstein distribution. Note that
can become very large in this distribution. The parameter is again determined
by the constraint on the total number of particles: i.e.,

(607) 
Equations (583) and (593) can be integrated to
give

(608) 
where use has been made of Eq. (606).
Note that photon statistics correspond to the special case of BoseEinstein
statistics in which the parameter takes the value zero, and the constraint
(607) does not apply.
Next: MaxwellBoltzmann statistics
Up: Quantum statistics
Previous: Photon statistics
Richard Fitzpatrick
20060202