Ultrarelativistic Electron Gas

Derive the relation between pressure and volume of a free ultrarelativistic electron gas at zero temperature.

SOLUTION

The fact that the gas is ultrarelativistic implies that the energy of the electron is large compared to its rest energy mc². In this case, the dispersion law is linear: ε = cp. The number of quantum states is the same as for the nonrelativistic case considered:

However, the Fermi energy now is different since ε = cp, and

Hence,

(1)

The total energy is

After substituting ε_F from (1), we obtain

So, the pressure is

Hence, for an ultrarelativistic gas we have PV = 3/E,  the same as for massless particles (e.g., photons), which is not surprising since the dispersion law is the same.