Crossover time of diffusion-limited reactions on a tubular lattice

Abstract

We studied the diffusion-limited reactions A+A\ensuremath{\rightarrow}0 and A+B\ensuremath{\rightarrow}0 and the number of distinct sites visited by a random walker on a d-dimensional tubular lattice: square lattice of sizes L\ifmmode\times\else\texttimes\fi{}${\mathrm{W}}^{\mathrm{d}\mathrm{\ensuremath{-}}1}$ with L\ensuremath{\gg}:W. We are interested in the crossover time at which the system changes its behavior from that in high dimensions to that in one dimension. We analytically solved the random-walk problem on the tubular lattice. Our theoretical result agrees with the simulation and thus explains the anomalous scaling of the crossover time for the random-walk problem. We also understood, using the concept of depletion zone, the scaling behavior of the crossover time for the reaction A+A\ensuremath{\rightarrow}0 on the tubular lattice. Our measurement and data collapse showed that the crossover time for the reaction A+B\ensuremath{\rightarrow}0 scales as ${\mathrm{W}}^{2}$ for large W. The discrepancy between our result and that of others is also discussed.