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

الأصل · EN

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⁻θ, where θ is the persistence exponent (``type I persistence''). We argue that, for a density of particles ρ>> 1, this non-trivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, where θ≈ 0.1207. In the case of an initially low density, ρ₀ << 1, we find θ≈ 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) (-const ρ₀¹/²t¹/⁴), provided ρ₀ << 1. A heuristic argument for this behavior, based on an exactly solvable toy model, is presented.