First of all we assume that we are dealing with a weakly quantum gas, i.e. . The system is a gran-canonical ensemble. Hence the partition function is:
Which is equivalent to:
Where the first expression holds for fermions, while the second one for bosons. In a gran canonical ensemble, the thermodynamic potential is given by the Landau thermodynamic potential:
Since it can be proved that , we also have: . From where one can derive the equation of state for weakly quantum gases:
Where the plus sign is for fermions, while the minus sign is for bosons.
Electrons in a metal can be idealized as fermions (electrons) in a box. Electrons will occupy energy levels according to the Fermi-Dirac distribution. In particular we have to note that:
We do define Fermi energy, the maximum of the energy of the occupied level: . If we define as the number of states, per unit volume, with energy in the interval we have:
The explicit expression of can be derived, obtaining:
Meaning that the Fermi energy equals to:
Hence we have that . The chemical potential depends on the temperature with the relation:
The specific heat, at constant volume , is defined as:
We now give three important relations that tell us what happens to the system when it happens a change in some quantity that describes it:
- in a canonical ensemble;
- in an isobaric ensemble;
- in a gran-canonical ensemble.
Quantum superfluids can be idealized as bosons in a box. Bosons in a box occupy energy levels according to the Bose-Einstein distribution. Moreover, if we do assume that all the particles occupy the level at , i.e. the lowes energy level, we have:
Where we do expect that , that is . Here we are assuming that a macroscopic number of particles occupy a certain energy level, more precisely the lowest possible! Actually, this is not a problem since bosons do not have to respect Pauli's exclusion principle. Meaning that the particles density is given by:
Where the contribute of is foundamental for temperatures , the Bose-Einstein condensation temperature. We also have that:
The condensation temperature can be derived and its value is:
And the condition is equivalent to:
Bose-Einstein condensation is a necessary but not sufficient condition to the superfluidity. For such systems we can define the specific heat, at constant pressure : . We have that:
- ↑ From and , using
- ↑ For a gas, given its density , its mass number (expressed in u.a.m) and its atomic number we can write .
- ↑ We can also note that:
- ↑ This condition is called Bose-Einstein condition.
- ↑ From