Transport in ultradilute solutions of 3He in superfluid 4He

We calculate the effect of a heat current on transporting He-3 dissolved in superfluid He-4 at ultralow concentration, as will be utilized in a proposed experimental search for the electric dipole moment of the neutron (nEDM). In this experiment, a phonon wind will be generated to drive (partly depolarized) 3He down a long pipe. In the regime of 3He concentrations less than or similar to 10(-9) and temperatures similar to 0.5 K, the phonons comprising the heat current are kept in a flowing local equilibrium by small angle phonon-phonon scattering, while they transfer momentum to the walls via the He-4 first viscosity. On the other hand, the phonon wind drives the 3He out of local equilibrium via phonon-He-3 scattering. For temperatures below 0.5 K, both the phonon and He-3 mean free paths can reach the centimeter scale, and we calculate the effects on the transport coefficients. We derive the relevant transport coefficients, the phonon thermal conductivity, and the He-3 diffusion constants from the Boltzmann equation. We calculate the effect of scattering from the walls of the pipe and show that it may be characterized by the average distance from points inside the pipe to the walls. The temporal evolution of the spatial distribution of the He-3 atoms is determined by the time dependent He-3 diffusion equation, which describes the competition between advection by the phonon wind and He-3 diffusion. As a consequence of the thermal diffusivity being small compared with the He-3 diffusivity, the scale height of the final He-3 distribution is much smaller than that of the temperature gradient. We present exact solutions of the time dependent temperature and He-3 distributions in terms of a complete set of normal modes.