Persistence in the One-Dimensional A+B -> 0 Reaction-Diffusion Model

Abstract

The persistence properties of a set of random walkers obeying the A+B -> 0 reaction, with equal initial density of particles and homogeneous initial conditions, is studied using two definitions of persistence. The probability, P(t), that an annihilation process has not occurred at a given site has the asymptotic form $P(t)->const+t^{-\theta }$, where $\theta$ is the persistence exponent (``type I persistence''). We argue that, for a density of particles $\rho >>1$, this non-trivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, where $\theta \approx 0.1207$. In the case of an initially low density, ${\rho }_0<<1$, we find $\theta \approx 1/4$ asymptotically. The probability that a site remains unvisited by any random walker (``type II persistence'') is also investigated and found to decay with a stretched exponential form, $P(t)\sim \exp (-const{\rho }_0^{1/2}{t}^{1/4})$, provided ${\rho }_0<<1$. A heuristic argument for this behavior, based on an exactly solvable toy model, is presented.

11 RevTeX pages, 19 EPS figures

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