Effect of Spatial Concentration Fluctuations on Effective Kinetics in Diffusion-Reaction Systems
The effect of spatial concentration fluctuations on reaction of two solutes, A+B$\rightarrow$C, is considered. In the absence of fluctuations, concentration of solutes decays as $A_{det}=B_{det} \sim t^{-1}$. Contrary to this, experimental and numerical studies suggest that concentrations decay significantly slower. Existing theory suggests $t^{-d/4}$ scaling in the asymptotic regime ($d$ is the dimensionality of the problem). Here we study the effect of fluctuations using the classical diffusion-reaction equation with random initial conditions. Initial concentrations of the reactants are treated as correlated random fields. We use the method of moment equations to solve the resulting stochastic diffusion-reaction equation and obtain a solution for the average concentrations that deviates from $\sim t^{-1}$ to $\sim t^{-d/4}$ behavior at time $t^*$. Finally, we derive analytical expressions for the time $t^*$ as a function of Damk\"{o}hler number and the coefficient of variation of the initial concentration. Our analytical results support earlier explanations of the deviation of average concentrations from $A_{det}=B_{det} \sim t^{-1}$ behavior, which attributed this change of scaling to the creation of isolated {\it islands} of A and B.