  For a potential function (in one dimension) which evolves from a specified
initial form $V_{i}(x)$ to a different $V_{f}(x)$ asymptotically, we study the
evolution, in an overdamped dynamics, of an initial probability density to its
final equilibeium.There can be unexpected effects that can arise from the time
dependence. We choose a time variation of the form
$V(x,t)=V_{f}(x)+(V_{i}-V_{f})e^{-\lambda t}$. For a $V_{f}(x)$, which is
double welled and a $V_{i}(x)$ which is simple harmonic, we show that, in
particular, if the evolution is adiabatic, the results in a decrease in the
Kramers time characteristics of $V_{f}(x)$. Thus the time dependence makes
diffusion over a barrier more efficient. There can also be interesting
resonance effects when $V_{i}(x)$ and $V_{f}(x)$ are two harmonic potentials
displaced with respect to each other that arise from the coincidence of the
intrinsic time scale characterising the potential variation and the Kramers
time.
