Size-Dependent Transport-Coefficient Effects in Fermi Liquids

Landau's theory of a Fermi liquid is applied to investigate the size-limited transport coefficients of a Fermi liquid contained in a long narrow channel of diameter $d$ small compared with the interquasiparticle mean free path $\lambda$. By characterizing the scattering of quasiparticles (QP) from the walls of the channel in a phenomenological way, expressions are obtained for the coefficients of spin diffusion, $D$, and of thermal conductivity, K. These expressions show, not unexpectedly, that D is a constant independent of $T$, giving a direct measure of the QP group velocity ${\mathrm{v}}_0$, while $K$ is proportional to $T$ and coincides with the result that would be obtained if we had considered a noninteracting Fermi gas. Mass flow through the channel, which for $d\ll \lambda$ is not characteristic of viscous flow, is also considered. Under the action of an externally applied pressure gradient, it is found that mass is discharged at a temperature-independent rate $G$ proportional to $d^3$: in the viscous regime ($d\gg \lambda$) $G$ varies as $d^4$. The expression obtained for $G$ is actually the Fermi-gas analog of the classical result originally obtained by Knudsen. The extension of the results to the regime $d\sim \lambda$ if is briefly considered. As a consequence of competition between Knudsen flow and Poiseuille flow, a "Knudsen minimum" will appear in the temperature dependence of $G$. Some numerical estimates are made for liquid ${\mathrm{He}}^3$.