Memory-Controlled Annihilation Reactions

We consider a diffusion-limited reaction in case the reacting entities are not available simultaneously. Due to the fact that the reaction takes place after a spatiotemporal accumulation of reactants, the underlying rate equation has to be modified by additional non-local terms. Owing to the delay effects a finite amount of reactants remains localized, preventing a further reaction and the asymptotic decay is terminated at a finite density. The resulting inhomogeneous non-zero stationary concentration is stable against long wave length fluctuations. Below a critical wave vector $k_c$ the system becomes inert, whereas a complete decay is realized above $k_c$. The phase diagram for the one species-annihilation process $A + A \to 0$ exhibits a behavior comparable to a second order phase transition. Obviously the memory effects are equivalent to long range interaction and the non-local kinetics is basically independent on space dimensions.