  The chemical master equation (CME) is the exact mathematical formulation of
chemical reactions occurring in a dilute and well-mixed volume. The
reaction-diffusion master equation (RDME) is a stochastic description of
reaction-diffusion processes on a spatial lattice, assuming well-mixing only on
the length scale of the lattice. It is clear that, for the sake of consistency,
the solution of the RDME of a chemical system should converge to the solution
of the CME of the same system in the limit of fast diffusion: indeed, this has
been tacitly assumed in most literature concerning the RDME. We show that, in
the limit of fast diffusion, the RDME indeed converges to a master equation,
but not necessarily the CME. We introduce a class of propensity functions, such
that if the RDME has propensities exclusively of this class then the RDME
converges to the CME of the same system; while if the RDME has propensities not
in this class then convergence is not guaranteed. These are revealed to be
elementary and non-elementary propensities respectively. We also show that
independent of the type of propensity, the RDME converges to the CME in the
simultaneous limit of fast diffusion and large volumes. We illustrate our
results with some simple example systems, and argue that the RDME cannot be an
accurate description of systems with non-elementary rates.
