Transport in ultradilute solutions of ³He in superfluid ⁴He

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We calculate the effect of a heat current on transporting ³He dissolved in superfluid ⁴He 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 generated to drive (partly depolarized) ³He down a long pipe. In the regime of ³He concentrations < 10⁻⁹ and temperatures ~ 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 ⁴He first viscosity. On the other hand, the phonon wind drives the ³He out of local equilibrium via phonon-³He scattering. For temperatures below $0.5$ K, both the phonon and ³He 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 ³He 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 ³He atoms is determined by the time dependent ³He diffusion equation, which describes the competition between advection by the phonon wind and ³He diffusion. As a consequence of the thermal diffusivity being small compared with the ³He diffusivity, the scale height of the final ³He distribution is much smaller than that of the temperature gradient. We present exact solutions of the time dependent temperature and ³He distributions in terms of a complete set of normal modes.