## Physics Friday 91

When we modeled the ideal Fermi fluid (previous posts here, here, here, here, and here), we used the fact that the grand canonical partition function can be factored over the individual states:
,
where the individual state partition is
,
where the sum is over all possible values of n, the number of particles in the state. For fermions, the Pauli exclusion principle meant that n=0 and n=1 were the only allowed values, and the above series had only two terms. Bosons, however, can be placed in the same quantum state in any number, so for an ideal Bose gas, the above sum becomes an infinite (geometric) series:
.

Next, let us find the occupancy (mean occupation number) fk,ms:
.
Now, we note that for |x|<1,
.
This tells us that the sum in the numerator is thus , and so
.
Compare this to the occupancy for a fermion fluid ; the difference is a change of sign in the denominator. Note that unlike the fermion case, for a boson fluid, the occupancy can be greater than one.

For a boson fluid, we generally choose our energy scale so that our zero energy is the ground state. Now, note that the occupancy has a singularity at , where it goes to infinity. This means that for a Bose gas with a finite number of particles, the Gibbs potential μ must always be less that all our energy values; thus, with the above choice of energy scale, we see that μ must always be negative.

We can, as in the fermion case, use the quantum particle in a cubical box to develop the concept of the “density of states” D(ε):
,
which allows us to approximate the sum over all states as an integral over energy:

(where g0 is the number of different spin states: g0=2s+1, where s is the particle spin, here an integer). Using the same integration by parts as in the fermion case, we can change this to:
.
Similarly,
,
and
.
Note the difference in signs in the denominators in the integrals from the fermion case, and that  still holds.
Using the math from this addendum, we can rewrite the above integrals in terms of polylogarithms:

and
.

We also note from our equation for Ñ, that if the number of particles is held constant, we see that the integral  must be constant, and so as the temperature increases, the GIbbs potential μ must decrease (as it also does for a Fermi gas).

When examining the ideal Fermi fluid in the classical limit, we noted that when the fugacity  is small (), we see that , and so the fermion occupancy can be approximated via
.
Now, note that  means that for the boson occupancy,
,
the exact same approximation. Thus, the results from applying this approximation is the same (except for the different value of g0), and thus, the ideal Bose gas has the same condition for classical behavior, and, as expected, also approaches the classical ideal gas. Thus, the difference between Bose gases and Fermi gases is only significant in the quantum regime.