In Diamond's OLG model, why is capital equal to saving by the young in previous period? In Diamond's OLG model, why is capital in the next period equal to total savings by the young in the current period?

What happeend to the pre-existing capital? Surely it doesn't just cease to exist?
I know it belongs to the old who die, but just because they die doesn't mean the capital disappears?
Property rights and assuming away bequest motives (the Old do no care about the Young).  

Capital of the previous period belongs to the Old, and since they will die at the and of old age, the finite-horizon transversality condition for utility maximization requires that either all assets have been consumed at the time of death, or that they have no utility-enhancing potential. Since the utility function reflects the "more is better" property, assets do have value as consumption down to the last bolt, and since there is no bequest motive, no utility from "leaving something for your children", it follows that the Old will consume all their capital, being utility maximizers.