D.61 Chemical potential in the distributions

The following convoluted derivation of the distribution functions comes fairly straightly from Baierlein [4, pp. 170-]. Let it not deter you from reading the rest of this otherwise very clearly written and engaging little book. Even a nonengineering author should be allowed one mistake.

The derivations of the Maxwell-Boltzmann, Fermi-Dirac, and
Bose-Einstein distributions given previously, {D.57}
and {D.58}, were based on finding the most numerous or
most probable distribution. That implicitly assumes that significant
deviations from the most numerous/probable distributions will be so
rare that they can be ignored. This note will bypass the need for
such an assumption since it will directly derive the actual
expectation values of the single-particle state occupation numbers

The mission is to derive the expectation number

where

Note that

It has the big consequence that the sum over the eigenfunctions can be replaced by sums over all sets of occupation numbers:

Each set of single-particle state occupation numbers corresponds to exactly one eigenfunction, so each eigenfunction is still counted exactly once. Of course, the occupation numbers do have to add up to the correct number of particles in the system.

Now consider first the case of

One simplification that is immediately evident is that all the terms that have

This can be simplified by taking the constant part of the exponential out of the summation. Also, the constraint in the bottom shows that the occupation numbers can no longer be any larger than

The right hand side falls apart into two sums: one for the 1 in

This equation is exact, no approximations have been made yet.

The system with

But when the system is macroscopic, the occupation counts for

The final formula is the Bose-Einstein distribution with

Solve for

The final fraction is a difference quotient approximation for the derivative of the Helmholtz free energy with respect to the number of particles. Now a single particle change is an extremely small change in the number of particles, so the difference quotient will be to very great accuracy be equal to the derivative of the Helmholtz free energy with respect to the number of particles. And as noted earlier, in the obtained expressions, volume and temperature are held constant. So,

Now consider the case of

Again, all terms with

But now there is a difference: even for a system with

Here are some additional remarks about the only approximation made,
that the systems with

But for bosons, it is a bit trickier because of the possibility of
condensation. Assume, reasonably, that when a particle is added, the
occupation numbers will not go down. Then the derived expression
overestimates both expectation occupation numbers

If the factor