On the reactions $A+A+\ldots+A\to 0$ at a one-dimensional periodic lattice of catalytic centers: Exact solution

The kinetics of the diffusion-controlled chemical reactions $A+A+\ldots+A\to 0$ that occur at catalytic centers periodically arranged along a straight line is considered. Modes of the behavior of reaction probability $W(t)$ were studied at different times and different densities of the catalyst. Within the Smoluchowski approximation, it was rigorously proved that at large times the function $W(t)$ is independent of the lattice period. This means that, in the given asymptotic mode, the probability of the reaction on a lattice with a catalyst placed in each lattice site is the same as on a lattice with a catalyst placed in sparse sites.