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We study the asymptotic and pre-asymptotic diffusive properties of Brownianparticles in channels whose section varies periodically in space. The effectivediffusion coefficient $D_{\mathrm{eff}}$ is numerically determined by theasymptotic behavior of the root mean square displacement in differentgeometries, considering even cases of steep variations of the channelboundaries. Moreover, we compared the numerical results to the predictions fromthe various corrections proposed in the literature to the well knownFick-Jacobs approximation. Building an effective one dimensional equation forthe longitudinal diffusion, we obtain an approximation for the effectivediffusion coefficient. Such a result goes beyond a perturbation approach, andit is in good agreement with the actual values obtained by the numericalsimulations. We discuss also the pre-asymptotic diffusion which is observed upto a crossover time whose value, in the presence of strong spatial variation ofthe channel cross section, can be very large. In addition, we show how theEinstein's relation between the mean drift induced by a small external fieldand the mean square displacement of the unperturbed system is valid in bothasymptotic and pre-asymptotic regimes.

Transport and fluctuation-dissipation relations in asymptotic and pre-asymptotic diffusion across channels with variable section