  We study the asymptotic and pre-asymptotic diffusive properties of Brownian
particles in channels whose section varies periodically in space. The effective
diffusion coefficient $D_{\mathrm{eff}}$ is numerically determined by the
asymptotic behavior of the root mean square displacement in different
geometries, considering even cases of steep variations of the channel
boundaries. Moreover, we compared the numerical results to the predictions from
the various corrections proposed in the literature to the well known
Fick-Jacobs approximation. Building an effective one dimensional equation for
the longitudinal diffusion, we obtain an approximation for the effective
diffusion coefficient. Such a result goes beyond a perturbation approach, and
it is in good agreement with the actual values obtained by the numerical
simulations. We discuss also the pre-asymptotic diffusion which is observed up
to a crossover time whose value, in the presence of strong spatial variation of
the channel cross section, can be very large. In addition, we show how the
Einstein's relation between the mean drift induced by a small external field
and the mean square displacement of the unperturbed system is valid in both
asymptotic and pre-asymptotic regimes.
