We derive and solve the Boltzmann equation for viscosity and diffusive thermal conductivity at low temperatures in the $B$ phase of superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The viscosity $\eta$ is shown to tend towards a constant value as the temperature tends to zero, with the constant being inversely proportional to an angular average of the collision probability. A general expression for the collision probability valid at any temperature is given in terms of the singlet and triplet components of the normal-state scattering amplitude. If one takes for the normal-state amplitude the $s$- and $p$-wave approximation, the constant viscosity is found to equal about one third of its value at the transition temperature. The diffusive thermal conductivity ${\kappa }_D$ is found to vary as $T^{-1\mathrm{}}$, as in the normal state, and with roughly the same coefficient of proportionality. We calculate as a function of pressure the viscosity and diffusive thermal conductivity in the normal state and in the superfluid at $T=0\mathrm{}$, and the normal-state quasiparticle relaxation time at the Fermi energy. The results are compared with experimental data, and the adequacy of the $s$- and $p$-wave approximation for the normal-state scattering amplitude is discussed. Finite temperature corrections to $\eta$ and ${\kappa }_{D}T$ are obtained for a particularly simple normal-state scattering amplitude, showing that $\eta$ initially decreases with increasing temperature while ${\kappa }_{D}T$ increases.

• Received 21 October 1976

DOI:https://doi.org/10.1103/PhysRevB.15.3384