Roton second sound and roton scattering

The theory of roton second sound in superfluid helium-4 is considered. We first derive the hydrodynamic equations which determine the motion of a pure roton gas. The collisions between rotons are assumed to conserve energy, momentum, and the number of rotons. The hydrodynamic equations without dissipative terms have the same form as the corresponding equations for an ordinary fluid. When lowest-order dissipative effects are included, it is found that there are some extra dissipative processes in addition to heat conduction and viscosity. We next consider modifications of the hydrodynamic equations which occur when the effects of collisions between rotons and phonons are included, and when there are some roton-roton collisions in which the roton number changes. It is found that at the temperatures and pressures where roton second sound has so far been observed, the wave motion is best described as isothermal second sound. We show that current experimental data indicate that in nearly all collisions between rotons (> 99.97%) the total number of rotons is conserved.