Decay kinetics of ballistic annihilation

Abstract

We study the kinetics of ballistic annihilation, A+A\ensuremath{\rightarrow}0, with continuous initial particle velocity distributions. The concentration and the rms velocity are found to decay as c\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ and ${\mathit{v}}_{\mathrm{rms}}$\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$, respectively, with the relation \ensuremath{\alpha}+\ensuremath{\beta}=1 holding in any spatial dimension. A ``mean-field'' Boltzmann equation for the evolution of the velocity distribution predicts that \ensuremath{\alpha} and \ensuremath{\beta} depend strongly on the initial condition, a behavior which is confirmed numerically in one and two dimensions.