Phonon number relaxation in a 3D superfluid with a concave acoustic branch

We consider the collisional evolution towards equilibrium of a spatially homogeneous and isotropic phonon gas of a three-dimensional superfluid with a concave acoustic excitation branch, at a non-zero but arbitrarily low temperature $T$. Three-phonon collisions $1\phi\leftrightarrow 2\phi$ are forbidden by conservation of energy-momentum. Four-phonon collisions $2\phi\to 2\phi$ of Landau and Khalatnikov lead, after a time $\propto T^{-7}$, only to a partial thermal equilibrium, a Bose law of non-zero chemical potential for the phonons, because they conserve the total number of phonons. Relaxation towards complete thermochemical equilibrium is therefore ensured by the much slower five-phonon collisions $2\phi\leftrightarrow 3\phi$ of Khalatnikov, in a time $\propto T^{-9}$. Using kinetic equations on the occupation numbers of the phonon modes and explicitly calculating the $2\phi\to 3\phi$ collisional amplitude with quantum hydrodynamics at low temperature, we determine the corresponding evolution of the fugacity $z_\phi$ of the phonon gas from the non-degenerate regime $z_\phi=0^+$ to complete equilibrium $z_\phi=1^-$. Using the conservation of total energy, we find that the fugacity varies with a non-integer power law $\propto t^{4/5}$ at short times and an exponential law at long times; the speed of change of entropy, always positive, is asymptotically proportional to the square of the speed of change of fugacity, $(\mathrm{d}/\mathrm{d}t)S_\phi\propto[(\mathrm{d}/\mathrm{d}t)z_\phi]^2$, as Landau predicted for an arbitrarily slow adiabatic transformation. Our results bring to a close the study initiated by Khalatnikov in 1950 and could be experimentally verified in a gas of cold fermionic atoms on the BCS side of the BEC-BCS crossover, or in superfluid liquid helium-4 at sufficiently high pressure.