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Using operator semigroup methods, we show that Fokker-Planck typesecond-order PDE-s can be used to approximate the evolution of the distributionof a one-step process on $N$ particles governed by a large system of ODEs. Theerror bound is shown to be of order $O(1/N)$, surpassing earlier results thatyielded this order for the error only for the expected value of the process,through mean-field approximations. We also present some conjectures showingthat the methods used have the potential to yield even stronger bounds, up to$O(1/N^3)$.

On the error of Fokker-Planck approximations of some one-step density dependent processes