  We derive a stochastic partial differential equation that describes the
fluctuating behaviour of reaction-diffusion systems of N particles, undergoing
Markovian, unary reactions. This generalises the work of Dean [J. Phys. A:
Math. and Gen., 29 (24), L613, (1996)] through the inclusion of random Poisson
fields. Our approach is based on weak interactions, which has the dual benefit
that the resulting equations asymptotically converge (in the N to infinity
limit) on a variation of a McKean- Vlasov diffusion, whilst still being related
to the case of Dean-like strong interactions via a trivial rescaling. Various
examples are presented, alongside a discussion of possible extensions to more
complicated reaction schemes.
