In the diaspora, where two days of yomim tovim are celebrated, the second day is typically an exact repetition of the first day. We say the same prayers, make a second Seder on Pesach, make kiddush ...
Out of Eretz Yisroel there are those that do Hakafos on Shmini Atzeres at night and some that even do it by day. Mostly I think it is by the Chasidim. Does anyone know of any Yeshivish or non ...