  A Langevin process diffusing in a periodic potential landscape has a time
dependent diffusion constant which means that its average mean squared
displacement (MSD) only becomes linear at late times. The long time, or
effective diffusion constant, can be estimated from the slope of a linear fit
of the MSD at late times. Due to the cross over between a short time
microscopic diffusion constant, which is independent of the potential, to the
effective late time diffusion constant, a linear fit of the MSD will not in
general pass through the origin and will have a non-zero constant term. Here we
address how to compute the constant term and provide explicit results for
Brownian particles in one dimension in periodic potentials. We show that the
constant is always positive and that at low temperatures it depends on the
curvature of the minimum of the potential. For comparison we also consider the
same question for the simpler problem of a symmetric continuous time random
walk in discrete space. Here the constant can be positive or negative and can
be used to determine the variance of the hopping time distribution.
