Consider a box of volume $V$ containing electron-positron pairs and photons in equilibrium at a temperature $T=1 / k \beta$. Assume that the equilibrium is established by the reaction
$$
\gamma \mapsto \mathrm{e}^{+}+\mathrm{e}^{-} .
$$
The reaction does not occur in free space, but one may think of it as catalyzed by the walls of the box. Ignoring the walls except insofar as they allow the reaction to occur, find
(a) The chemical potentials for the fermions.
(b) The average number of electron-positron pairs, in the two limits $k T \gg m_e c^2$ and $k T \ll m_e c^2$. (You may leave your answers in terms of dimensionless definite integrals.)
(c) The neglect of the walls is not strictly permissible if they contain a matter-antimatter imbalance. Supposing that this imbalance creates a net chemical potential $\mu \neq 0$ for the electrons, what is then the chemical potential of the positrons?
(d) Calculate the net charge of the system in the presence of this imbalance in the limit $k T \gg \mu \gg m_{\mathrm{e}} c^2$. (Again, your answer may be left in terms of a dimensionless definite integral.)