# Equation-of-state for a photon gas

What is the equation-of-state for the photon gas ?

Remarks:

• Check Wikipedia: photon gas Commented Sep 22, 2023 at 9:36
• @Mauricio thanks, this is actually what I was looking for. Commented Sep 22, 2023 at 9:38

The simplest case is when the radiation fluid is modelled as an ideal ultrarelativistic gas. In this case, the relation $$P=\epsilon/3$$, where $$\epsilon$$ is the energy density, is the equation of state for the photon component of the multifluid: this equation of state is valid for any ideal ultrarelativistic gas, both Bosonic or Fermionic (i.e. it is valid as long as we are in a limit where we can ignore mass or in a genuinely massless case).
In fact, $$P=\epsilon/3$$ is even more general: it can be used as an equation of state, but also as a kinematic identity (it is valid also out of thermodynamic equilibrium because it is a requirement stemming from the fact that the energy-momentum tensor of photons is traceless!). The real difference between the equilibrium and non-equilibrium case is the fact that at equilibrium (i.e. in the black-body case, see this answer for the difference with the non-equilibrium case) we know that the energy density is proportional to the 4th power of the temperature: $$\epsilon \propto p \propto T^4$$, see e.g. this question.