  The role of dimensionality (Euclidean versus fractal), spatial extent,
boundary effects and system topology on the efficiency of diffusion-reaction
processes involving two simultaneously-diffusing reactants is analyzed. We
present numerically-exact values for the mean time to reaction, as gauged by
the mean walklength before reactive encounter, obtained via application of the
theory of finite Markov processes, and via Monte Carlo simulation. As a general
rule, we conclude that for sufficiently large systems, the efficiency of
diffusion-reaction processes involving two synchronously diffusing reactants
(two-walker case) relative to processes in which one reactant of a pair is
anchored at some point in the reaction space (one walker plus trap case) is
higher, and is enhanced the lower the dimensionality of the system. This
differential efficiency becomes larger with increasing system size and, for
periodic systems, its asymptotic value may depend on the parity of the lattice.
Imposing confining boundaries on the system enhances the differential
efficiency relative to the periodic case, while decreasing the absolute
efficiencies of both two-walker and one walker plus trap processes. Analytic
arguments are presented to provide a rationale for the results obtained. The
insights afforded by the analysis to the design of heterogeneous catalyst
systems are also discussed.
