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FermiDirac statistics
Let us, first of all, consider FermiDirac statistics. According to Eq. (582),
the average number of particles in quantum state can be written

(584) 
Here, we have rearranged the order of summation, using the multiplicative
properties of the exponential function. Note that the first sums in the
numerator and denominator
only involve , whereas the last sums
omit the particular state from consideration
(this is indicated by the superscript on the summation symbol). Of course,
the sums in the above expression range over all values of the
numbers
such that and 1 for
each , subject to the overall constraint that

(585) 
Let us introduce the function

(586) 
which is defined as the partition function for particles distributed over all
quantum states, excluding state , according to FermiDirac statistics.
By explicitly performing the sum over and 1, the
expression (584) reduces to

(587) 
which yields

(588) 
In order to make further progress, we must somehow relate to
. Suppose that
. It follows that
can be Taylor expanded to give

(589) 
where

(590) 
As always, we Taylor expand the slowly varying function , rather
than the rapidly varying function , because the radius of
convergence of the latter Taylor series is too small for the series to
be of any practical use. Equation (589) can be rearranged to give

(591) 
Now, since is a sum over very many
different quantum states, we would not expect the logarithm of this
function to be sensitive to which particular state
is excluded from consideration. Let us, therefore, introduce the
approximation that is independent of , so that we can write

(592) 
for all . It follows that the derivative (590) can be expressed
approximately in terms of the derivative of the full partition function
(in which the particles are distributed over
all quantum states). In fact,

(593) 
Making use of Eq. (591), with
, plus the
approximation (592), the expression (588) reduces to

(594) 
This is called the FermiDirac distribution. The parameter
is determined by the constraint that
: i.e.,

(595) 
Note that
if becomes sufficiently
large.
On the other hand, since the denominator in Eq. (594) can never become
less than unity, no matter how small becomes, it follows
that
. Thus,

(596) 
in accordance with the Pauli exclusion principle.
Equations (583) and (593) can be integrated to
give

(597) 
where use has been made of Eq. (594).
Next: Photon statistics
Up: Quantum statistics
Previous: Formulation of the statistical
Richard Fitzpatrick
20060202