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

Abstract

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