Let $g(p)$ be the density of states of a particle in volume $V$ as a function of the magnitude $p$ of the particle's momentum. Explain why $g(p) \propto V p^{2} / h^{3}$, where $h$ is Planck's constant. Write down the Bose-Einstein and Fermi-Dirac distributions for the (average) number $\bar{n}(p)$ of particles of an ideal gas with momentum $p$. Hence write down integrals for the (average) total number $N$ of particles and the (average) total energy $E$ as functions of temperature $T$ and chemical potential $\mu$. Why do $N$ and $E$ also depend on the volume $V ?$

Electromagnetic radiation in thermal equilibrium can be regarded as a gas of photons. Why are photons "ultra-relativistic" and how is photon momentum $p$ related to the frequency $\nu$ of the radiation? Why does a photon gas have zero chemical potential? Use your formula for $\bar{n}(p)$ to express the energy density $\varepsilon_{\gamma}$ of electromagnetic radiation in the form

$$\varepsilon_{\gamma}=\int_{0}^{\infty} \epsilon(\nu) d \nu$$

where $\epsilon(\nu)$ is a function of $\nu$ that you should determine up to a dimensionless multiplicative constant. Show that $\epsilon(\nu)$ is independent of $h$ when $k T \gg h \nu$, where $k$ is Boltzmann's constant. Let $\nu_{\text {peak }}$ be the value of $\nu$ at the maximum of the function $\epsilon(\nu)$; how does $\nu_{\text {peak }}$ depend on $T$ ?

Let $n_{\gamma}$ be the photon number density at temperature $T$. Show that $n_{\gamma} \propto T^{q}$ for some power $q$, which you should determine. Why is $n_{\gamma}$ unchanged as the volume $V$ is increased quasi-statically? How does $T$ depend on $V$ under these circumstances? Applying your result to the Cosmic Microwave Background Radiation (CMBR), deduce how the temperature $T_{\gamma}$ of the CMBR depends on the scale factor $a$ of the Universe. At a time when $T_{\gamma} \sim 3000 K$, the Universe underwent a transition from an earlier time at which it was opaque to a later time at which it was transparent. Explain briefly the reason for this transition and its relevance to the CMBR.

An ideal gas of fermions $f$ of mass $m$ is in equilibrium at temperature $T$ and chemical potential $\mu_{f}$ with a gas of its own anti-particles $\bar{f}$ and photons $(\gamma)$. Assuming that chemical equilibrium is maintained by the reaction

$$f+\bar{f} \leftrightarrow \gamma$$

determine the chemical potential $\mu_{\bar{f}}$ of the antiparticles. Let $n_{f}$ and $n_{\bar{f}}$ be the number densities of $f$ and $\bar{f}$, respectively. What will their values be for $k T \ll m c^{2}$ if $\mu_{f}=0$ ? Given that $\mu_{f}>0$, but $\mu_{f} \ll k T$, show that

$$n_{f} \approx n_{0}(T)\left[1+\frac{\mu_{f}}{k T} F\left(m c^{2} / k T\right)\right]$$

where $n_{0}(T)$ is the fermion number density at zero chemical potential and $F$ is a positive function of the dimensionless ratio $m c^{2} / k T$. What is $F$ when $k T \ll m c^{2}$ ?

Given that $\mu_{f} \ll k T$, obtain an expression for the ratio $\left(n_{f}-n_{\bar{f}}\right) / n_{0}$ in terms of $\mu, T$ and the function $F$. Supposing that $f$ is either a proton or neutron, why should you expect the ratio $\left(n_{f}-n_{\bar{f}}\right) / n_{\gamma}$ to remain constant as the Universe expands?