Annihilation kinetics in the one-dimensional ideal gas

Abstract

We consider the annihilation reaction A+A\ensuremath{\rightarrow}0 on the real line with an initial equilibrium distribution of particles A (points or rods); the particles move freely before annihilating, with independent initial velocities. For the dichotomic distribution of velocities P(v=c)=1-P(v=-c)=p, we determine the fraction S(t) of particles surviving at time t: (1) If p=(1/2), S(t)\ensuremath{\sim}${t}^{\mathrm{\ensuremath{-}}1/2}$ by a central-limit effect; (2) if p\ensuremath{\ne}(1/2), S(t)\ensuremath{\sim}\ensuremath{\Vert}2p-1\ensuremath{\Vert}+${\mathrm{At}}^{\mathrm{\ensuremath{-}}3/2}$exp(-Bt) with known constants A and B. We also determine the asymptotic spatial distribution of surviving particles. From these results we derive some bounds on the decay of S(t) for other velocity distributions, and we compare them to the decay of S(t) for diffusive (Brownian) motion.