  We calculate the probability distribution function (PDF) of an overdamped
Brownian particle moving in a periodic potential energy landscape $U(x)$. The
PDF is found by solving the corresponding Smoluchowski diffusion equation. We
derive the solution for any periodic even function $U(x)$, and demonstrate that
it is asymptotically (at large time $t$) correct up to terms decaying faster
than $\sim t^{-3/2}$. As part of the derivation, we also recover the
Lifson-Jackson formula for the effective diffusion coefficient of the dynamics.
The derived solution exhibits agreement with Langevin dynamics simulations when
(i) the periodic length is much larger than the ballistic length of the
dynamics, and (ii) when the potential barrier $\Delta U=\max(U(x))-\min(U(x))$
is not much larger than the thermal energy $k_BT$.
