Fluctuations and Order pp 171-203 | Cite as

# The Hierarchies of Nonclassical Regimes for Diffusion-Limited Binary Reactions

## Abstract

Diffusion-limited binary reactions in low dimensions may lead to the spontaneous formation of spatial structures and to associated “anomalous” rate laws for the global densities of the reacting species. For example, the irreversible reactions *A* + *A* → 0 and *A* + *B* → 0 under “normal” circumstances are described by second-order rate laws, whereas the asymptotic rate law for the former reaction is of apparent order (1 + 2/*d*) for dimensions *d* < 2 and for the mixed reaction it is of apparent order (1 + 4/*d*) for *d* < 4. The slowdown implied by the higher order is a consequence of the spatial distribution of reactants and its rapid deviation from a random distribution, which is in turn a consequence of the fact that diffusion is not an effective mixing mechanism in low dimensions. The principal effect in the mixed reaction is the formation of aggregates of like particles. The spatial regions in which the density of one type of particle is overwhelmingly greater than that of the other grow in time. Since the reaction can essentially occur only at the interfaces between aggregates, and since the number of these interfaces decreases with time, the reaction slows down relative to the rate that would describe a random mixture of reactants. Note that initial spatial fluctuations in relative densities are essential for this ordering effect to occur: These fluctuations grow in size as the reaction that eliminates close opposite pairs proceeds. The particular rate law of order (1 + 4/*d*) appropriate for an initial distribution of reactants that is completely random changes if the initial distribution is not totally random. In this chapter, we discuss the various regimes of kinetic behavior of the densities of reactants from the initial time until the asymptotic behavior is reached, and we estimate the crossover times from one regime to another. Our analysis deals with the effects of the initial conditions on this characterization. In particular, we find that initial spatial correlations limit the initial (and hence subsequent) fluctuations in the particle numbers, and hence they affect the rate laws and the underlying spatially segregated patterns. We also present numerical simulation results in one and two dimensions and analyze these results in terms of our model.

## Keywords

Initial Density Finite Size Effect Depletion Zone Kinetic Regime Random Initial Condition## Preview

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