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For the purpose of comparison, it is instructive to consider the purely
classical case of MaxwellBoltzmann statistics. The partition
function is written

(609) 
where the sum is over all distinct states of the gas, and the particles
are treated as distinguishable. For given values of
there
are

(610) 
possible ways in which distinguishable
particles can be put into individual quantum states such that there are
particles in state 1, particles in state 2, etc. Each of these possible
arrangements corresponds to a distinct state for the whole gas.
Hence, Eq. (609) can be written

(611) 
where the sum is over all values of
for each , subject to
the constraint that

(612) 
Now, Eq. (611) can be written

(613) 
which, by virtue of Eq. (612), is just the result of
expanding a polynomial. In fact,

(614) 
or

(615) 
Note that the argument of the logarithm is simply the partition function
for a single particle.
Equations (583) and (615) can be combined to give

(616) 
This is known as the MaxwellBoltzmann distribution. It is, of course,
just the result obtained by applying the Boltzmann distribution to a single
particle (see Sect. 7).
Next: Quantum statistics in the
Up: Quantum statistics
Previous: BoseEinstein statistics
Richard Fitzpatrick
20060202