Adiabatic elimination in stochastic systems. II. Application to reaction diffusion and hydrodynamic-like systems

We develop a stochastic theory of rapidly diffusing spatially distributed systems. Discussion is within the framework of the cell model, in which the system is described in terms of a lattice of $n$ cells. Utilizing projector-operator techniques, we formalize the method of homogenization of such systems. That is, by projecting out high-$q$ Fourier modes in the adiabatic limit of large diffusion, we map the system to one defined on a coarser-grained lattice. We thus demonstrate a "blocking" procedure in the cell model. Finally we consider a simple hydrodynamic model and show that near the point of convective instability projection-operation methods predict the same amplitude equations for the slow hydrodynamic modes as does the method of multiple scales.