Overdamped modes and transport in normal ³He

In earlier work it was shown that a sinusoidal density distribution ~cos α x at t = 0 decays to a uniform state with the excitation of damped, standing waves of first sound, in the long-wavelength limit, plus overdamped solutions. Since the latter are independent solutions of the Boltzmann equation, we should expect that as α → 0, the heat flux J and temperature gradient ▽T in such a relaxing mode are related by Fourier's law, and indeed it is shown that explicit calculation of J and ▽T from the longest lived overdamped solution yields a theoretical expression relating the thermal conductivity λ to the Landau parameters F ₀ ^s, F ₁ ^s, and F ₂ ^s. To evaluate λ, one needs to calculate the internal energy per particle to second order in the departure from equilibrium of the quasi-particle momentum distribution, and this is effected with the aid of a reciprocity relation linking the pressure and the viscoelastic relaxation equation. In the final expression for λ, F ₂ ^s is adjusted to make the theoretical thermal conductivity fit the value tabulated by Wheatley, giving F ₂ ^s = −1.68, which agrees in sign and magnitude with previous estimates from longitudinal and transverse zero sound. To establish the physical significance of the relaxing solution, it is shown that its contribution to the f-sum rule, combined with the contribution from the first-sound poles of the dynamic response function, satisfies the sum rule exactly to fourth order in the wave number. Other relaxing modes contribute in higher order.