Kinetics ofn-species annihilation: Mean-field and diffusion-controlled limits

Abstract

We consider the kinetics of a system where n distinct chemical species $A_i$ undergo the reactions $A_i$+$A_j$→inert (i≠j). In the rate-equation approximation, a conservation law for the reaction is derived, and an explicit solution for the case n=3 is given. At the level of the master equation, Van Kampen’s Ω expansion is employed to estimate the magnitude of local fluctuations in density which arise when the basic dynamical variable of the system is the (discrete) particle number, rather than a continuous particle density. We then develop a physical picture for the evolution of the n-species system, when the particles diffuse, which, together with the estimate of the magnitude of the local fluctuations, is used to deduce the form of the decay law when the initial densities of the n species are equal. For a d-dimensional n-species system below an upper critical dimension equal to 4(n-1)/(2n-3), the density is predicted to decay as $t^{\mathrm{-}\alpha \left(n\right)}$, with α(n)=1/2d{1-1/[2(n-1)]}, and this is verified by numerical simulations. In addition, the simulations reveal a striking symmetry breaking, where the equal-density initial state evolves to a long-time state where one species predominates.