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