  The transition from the instanton-dominated quantum regime to the
sphaleron-dominated classical regime is studied in the $d=2$ abelian-Higgs
model when the spatial coordinate is compactified to $S^1$. Contrary to the
noncompactified case, this model allows both sharp first-order and smooth
second-order transitions depending on the size of the circle. This finding may
make the model a useful toy model for the analysis of baryon number violating
processes. Since the model can to a large extent be treated analytically, it
can also serve as a transparent prototype for the application of our method to
more complicated cases, such as those in higher dimensions.
