This is the second in the series of papers on transport phenomena alongrandom rough surfaces. We apply our simple general approach\cite{r1} totransport in very narrow channels, when the particles wavelength is comparableto the width of the channels and the motion across the channel is characterizedby discrete quantum states. The discrete nature of the spectrum leads to anon-analyticity of transport coefficients as a function of thickness or channelwidth, especially for degenerate fermions. Surface inhomogeneity leads to bothon-level scattering and interlevel transitions. As in\cite{r1}, transportcoefficients are expressed explicitly via the correlation function of surfaceinhomogeneities. The shape of the curves for the dependence of transportcoefficients on the number of particles and/or film thickness is determined bythe correlation radius of surface inhomogeneities and is not sensitive to theiramplitude. For short range correlations, the curves assume a saw-like shape asa result of the interlevel transitions. With increasing correlation radius, theinterlevel transitions loose their importance, and the saw teeth graduallydecrease, reducing, in the end, to simple kinks on practically monotoniccurves. Applications include Gaussian and short-range $\delta $-typecorrelations. Careful analysis of the transition from quantum to semi-classicaland classical regimes allowed us to improve the accuracy of our previousclassical calculations. PACS numbers: 72.10.Bg, 73.20.Fz, 73.50.Bk, 79.20.Rf
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