Decay Kinetics in Ballistic Annihilation

For irreversible diffusion-controlled reactions, it is now widely appreciated that the density decays more slowly than the predictions of mean-field theory in sufficiently low spatial dimension. For two-species annihilation, this behavior is accompanied by the dynamic formation of large-scale spatial heterogeneities in an initially homogeneous system [1]. The contrasting situation where the reactants move ballistically has received much less attention, however, and relatively little is known. A number of interesting results have been recently reported for the kinetics of irreversible aggregation, Ai + Aj → Ai+j , with ballistic trajectories for the aggregates and with momentum conserving collisions [2,3]. Here the subscript refers to the (conserved) mass of the aggregates. This model has been invoked as an idealization of processes such as the coalescence of fluid vortices [4] and planet formation by accretion [5]. For the ballistic aggregation model, a scaling argument suggests that the concentration of aggregates decays as t−α, with α = 2d/(d+2), where d is the spatial dimension [2]. This dimension dependence for all d is atypical of the behavior pattern exhibited by diffusion-controlled reactions. Furthermore, microscopic considerations show that the decay of the density of fixed-mass aggregates disagrees with the predictions of the scaling argument [3]. Motivated in part by these intriguing features, we investigate the decay kinetics of the more elementary single-species annihilation process, A + A → 0, for arbitrary continuous initial velocity distributions. We find that the decay of the density depends non-universally on the initial velocity distribution. A Boltzmann equation for the evolution of the velocity distribution accounts for the dependence of the decay exponent α on the form of the velocity distribution and on the spatial dimension. Our predictions are verified in one and two dimensions by numerical integration of the Boltzmann equation and by Monte Carlo simulations. It is worth noting that for one-dimensional single-species annihilation with a discrete bimodal initial velocity distribution, P (v, t = 0) ∝ pδ(v − 1) + (1 − p)δ(v + 1), the density decays as t−1/2 for p = 1/2, while the minority velocity species decays exponentially for p 6= 1/2 [6]. These results can be inferred by mapping the kinetics onto a first-passage process for a one-dimensional random walk. When the velocities are continuously distributed, this line of reasoning is inadequate to account for the wide range of possible kinetic behaviors. At time t = 0, the system consists of identical particles which are distributed in space with P (v, t)i the initial concentration of particles of velocity v. Without loss of generality the average initial velocity can be chosen to be zero. The decay kinetics appears to be independent of the initial spatial distribution of particles and for simplicity we focus on a random initial distribution. Particles move according to their initial velocity until a collision occurs, which results in the removal of both colliding particles. We are interested in determining the time dependence of the macroscopic concentration, c(t) = ∫ dvP (v, t), and the moments of the velocity distribution, 〈v〉 = ( ∫ dv vP (v, t)/c(t) )1/n. A simple power counting argument relates the density decay exponent α with the exponent β which characterizes the decay of the typical velocity, vrms ∼ t−β . Consider a system of identical particles of fixed radius r at concentration c which move with a velocity of the order of vrms. From an elementary mean-free path argument, the time between collisions is t ∼ 1/cvrmsr. If one assumes the following power law forms for the concentration and vrms, c ∼ t−α vrms ∼ t−β , (1)