Low Temperature Transport Properties of Very Dilute Classical Solutions of ${}^3$He in Superfluid ${}^4$He

We report microscopic calculations of the thermal conductivity, diffusion constant and thermal diffusion constant for classical solutions of ${}^3$He in superfluid ${}^4$He at temperatures $T\text{la}0.6$~K, where phonons are the dominant excitations of the ${}^4$He. We focus on solutions with ${}^3$He concentrations $\text{la}{10}^{-3}$, for which the main scattering mechanisms are phonon-phonon scattering via 3-phonon Landau and Beliaev processes, which maintain the phonons in a drifting equilibrium distribution, and the slower process of ${}^3$He-phonon scattering, which is crucial for determining the ${}^3$He distribution function in transport. We use the fact that the relative changes in the energy and momentum of a ${}^3$He atom in a collision with a phonon are small to derive a Fokker-Planck equation for the ${}^3$He distribution function, which we show has an analytical solution in terms of Sonine polynomials. We also calculate the corrections to the Fokker-Planck results for the transport coefficients.