  We examine the dispersion of Brownian particles in a symmetric two
dimensional channel, this classical problem has been widely studied in the
literature using the so called Fick-Jacobs' approximation and its various
improvements. Most studies rely on the reduction to an effective one
dimensional diffusion equation, here we drive an explicit formula for the
diffusion constant which avoids this reduction. Using this formula the
effective diffusion constant can be evaluated numerically without resorting to
Brownian simulations. In addition a perturbation theory can be developed in
$\varepsilon = h_0/L$ where $h_0$ is the characteristic channel height and $L$
the period. This perturbation theory confirms the results of Kalinay and Percus
(Phys. Rev. E 74, 041203 (2006)), based on the reduction, to one dimensional
diffusion are exact at least to ${\cal O}(\varepsilon^6)$. Furthermore, we show
how the Kalinay and Percus pseudo-linear approximation can be straightforwardly
recovered. The approach proposed here can also be exploited to yield exact
results an appropriate limit $\varepsilon \to \infty$, we show that here the
diffusion constant remains finite and show how the result can be obtained with
a simple physical argument. Moreover we show that the correction to the
effective diffusion constant is of order $1/\varepsilon$ and remarkably has a
some universal characteristics. Numerically we compare the analytical results
obtained with exact numerical calculations for a number of interesting channel
geometries.
