Roton second sound in the relaxation-time approximation

In the hydrodynamic limit the Boltzmann equation for rotons is solved in the relaxation-time approximation with two relaxation times. One of them characterizes the approach of the system to a local equilibrium with constant and zero chemical potential (the collisions between the elementary excitations do not conserve total number), whereas the other one scales the approach to a local equilibrium with nonvanishing chemical potential (total number conserved). The solution is used to investigate the velocity and the attenuation of second sound. The velocity is found to change as a function of the second-sound frequency from a value of $\frac{\sqrt{3}kT}{p_0}$ at frequencies which are slow compared to the frequency of number-nonconserving collisions to a value of $\frac{3kT}{p_0}$ in the opposite limit. $p_0$ is the momentum at the roton minimum. The corresponding attenuation displays a maximum in the region of steepest change of the velocity.