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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 $T \la 0.6$~K, where phonons are the dominant excitations of the $^4$He. We focus on solutions with $^3$He concentrations $\la \,10^{-3}$, 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.