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The transition from the instanton-dominated quantum regime to thesphaleron-dominated classical regime is studied in the $d=2$ abelian-Higgsmodel when the spatial coordinate is compactified to $S^1$. Contrary to thenoncompactified case, this model allows both sharp first-order and smoothsecond-order transitions depending on the size of the circle. This finding maymake the model a useful toy model for the analysis of baryon number violatingprocesses. Since the model can to a large extent be treated analytically, itcan also serve as a transparent prototype for the application of our method tomore complicated cases, such as those in higher dimensions.Comment: revtex, 16 pages, 4 figures. version to appear in International J. of  Modern Physics

INSTANTON-SPHALERON TRANSITION IN THE d=2 ABELIAN–HIGGS MODEL ON A CIRCLE