## Physics Friday 88

Previously, I used the grand canonical formalism to derive the fundamental thermodynamic relation for a system of spin-½ fermions in a quantum system with a small number of orbital states. Now, let us extend this analysis to the ideal Fermi fluid, the quantum analog to the classical ideal gas for fermions; a system of fermion particles with no (or negligible) interaction forces, such as a collection of neutrons in the interior of a neutron star, or a low-temperature gas of 3He atoms.
As with our previous work, we will use the grand canonical formalism; the fundamental relations will be independent of these particular boundary conditions. We consider spin-½ fermions, so that for each orbital state we have two states, given by ms=±½. The orbital states available can be specified by the wavenumber vector k of the wavefunction (see here). As before, our grand partition sum factors: . As we are considering fermions, each orbital state is either empty or singly occupied. The energy of an empty state is zero, and the energy of an occupied state (k,ms) is  (independent of the spin direction). This means the partition sum for a state (k,ms) is

In turn, we can consider the product of the two states of orbital k and opposite spin directions as the “partition sum of mode k, with terms for the empty mode, singly occupied mode (with two possible orientations), and doubly occupied mode.
Each state is independent, so we have mean occupation number

(in analogy to here).
Now, we can find the grand potential:
.

Now, consider our particle confined to a cubical box of volume V=L3; I demonstrated the quantum particle in a cubical box here; we have the allowed orbital states given by , , where nx, ny, and nz are positive integers.
Thus, the number of states of energy less than or equal to E is the number of triplets of positive integers (nx, ny, nz) for which , where . Considering the three dimensional abstract space with coordinates (nx, ny, nz), we see our points are those contained in the eighth of the sphere of radius n  in the first octant. As each point corresponds to a cube of unit volume in this space (visualize the lattice of integer points), for large n, we can approximate the number of orbital states of energy less than or equal to E by the volume of this eighth of the sphere:
.
Differentiating this with respect to energy, we can obtain a “density of (orbital) states”:
.
If we multiply this by two, since we have two spin orientations per each orbital state, we can approximate our sum in the grand potential with an integral:
.
There is no closed-form expression for the integral (though it can be expressed in terms of polylogarithms), so the physical properties derived from Ψ must also be expressed in terms of integrals (or polylogarithms), which can be computed numerically or via approximation methods.
We can, though, immediately determine the number of particles Ñ and internal energy U:
,
(which we could also have obtained via  and our integral expression for Ψ),
and
.

[These integrals may also be expressed in terms of polylogarithms:
,
and
. See here for more of the math involved.]

Now, if we perform integration by parts on the integral for Ψ, with  and , we obtain
.
We also recall from the end of this post, that for simple systems, the equation of state is . Thus, we see that
. Note that this is exactly the result for a classical ideal gas whose molecules have only translational degrees of freedom available (see here and here).

Next week, I intend to show that this reduces to the classical ideal gas for high temperature, and extract the criterion separating the quantum and classical regimes.