  The dynamics of a freely diffusing particle in a two-dimensional channel with
cross sectional area $A(x)$, can be effectively described by a one-dimensional
diffusion equation under the action of a potential of mean force $U(x)=-k_BT\ln
[A(x)]$ (where $k_BT$ is the thermal energy) in a system with a
spatially-dependent diffusion coefficient $D(x)$. Several attempts to derive
expressions relating $D(x)$ to $A(x)$ and its derivatives have been made, which
were based on considering stationary flows in periodic channels. Here, we take
an alternative approach and consider non-steady state single particle diffusion
in an open periodic channel. The approach allows us to express $D(x)$ as a
series of terms of increasing powers of $\epsilon$ - a parameter associated
with the aspect ratio of the channel. When the expansion is truncated at the
leading term, we recover the expression suggested by Zwanzig [J. Phys. Chem.
{\bf 96}, 3926 (1992)] for $D(x)$. Furthermore, comparison of the first few
terms in our expansion for $D(x)$ with the one proposed by Kalinay and Percus
[Phys. Rev. E {\bf 74}, 041203 (2006)] shows that they are consistent with each
other. In the limit of long wavelength channels ($\epsilon\ll 1$), the
expansion converges rapidly and the leading approximation provides a very
accurate description of the two-dimensional dynamics. For short wavelength
channels, the expansion does not converge and the validity of the effective
one-dimensional description is questionable.
