Reactive dynamics on fractal sets: anomalous fluctuations and memory effects

We study the effect of fractal initial conditions in closed reactive systems in the cases of both mobile and immobile reactants. For the reaction $A+A\to A$, in the absence of diffusion, the mean number of particles $A$ is shown to decay exponentially to a steady state which depends on the details of the initial conditions. The nature of this dependence is demonstrated both analytically and numerically. In contrast, when diffusion is incorporated, it is shown that the mean number of particles $<N(t)>$ decays asymptotically as $t^{-d_f/2}$, the memory of the initial conditions being now carried by the dynamical power law exponent. The latter is fully determined by the fractal dimension $d_f$ of the initial conditions.