Radiation pressure and the black-body

Let’s consider now a closed box of volume $V$containing a photon gaz at equilibrium at temperature $T$. The gaz is thermalized by the constant interactions with the electrons of the atoms of the walls and has a continus spectrum. (this spectrum would then be emitted by the gaz if a hole was open in it). Photons are continuiously created and absorbed and it’s impossible to fix their average value $N$ when minimizing the entropy (since they have no mass, one can get any state of energy $E$ with N photons of energy $\hbar \omega$ or 2N photons of energy $\hbar \frac{\omega}{2}$ $\cdots$). The number of photons $N$ has no impact on the macrostate which is then fully determined by $E$. One must then fix $\mu=0$.

\[\langle \epsilon_\lambda\rangle = \langle n_\lambda\rangle\epsilon_\lambda = \langle n_\omega \rangle \hbar \omega\] \[g(\omega)= \frac{V}{\pi^2 c^3}\omega^2\] \[u(\omega)= \frac{1}{V}\frac{dE}{d\omega}=\frac{1}{\pi^2c^3}\frac{\hbar \omega^3}{e^{\beta \hbar \omega}-1}\] \[B_\nu(T)= u(\omega)\frac{d\omega}{d\nu}\] \[w=\frac{1}{3}\]