# 8.10. Ideal Fermi Gas¶

We start with a grand potential for fermions and use a Thomas-Fermi approximation (that allows us to change the discrete sum below into a continuous integral):

Note: to write this thermodynamic potential in the canonical form , we simply use the relation and get:

Let us compute the particle density:

and express the chemical potential as a function of the particle density :

(8.10.1)

We write the grand potential using as follows:

(8.10.2)

Now we can calculate the free energy:

where we used (8.10.2), (8.10.1) and the fact that . Note: we can express the free energy in canonical form using and :

We can calculate the entropy as follows:

The total energy is then equal to:

Note: the kinetic energy is equal to the total energy, as the gas is non-interacting.

The pressure can be calculated from:

Note that we got , , and .

## 8.10.1. Low Temperature Limit¶

At low temperature () we have , (for ) and we obtain:

Identical with the zero temperature Thomas-Fermi equation where the chemical potential becomes the Fermi energy in the limit . We now express in terms of at :

and compute pressure at using for :

## 8.10.2. High Temperature Limit¶

At high temperature () we have , (for ) and we obtain:

We now express in terms of at :

In the limit we get . Let us compute pressure at using for :

We obtained the ideal gas equation .