Two scales in asynchronous ballistic annihilation

The kinetics of single-species annihilation, A + A → 0, is investigated in which each particle has a fixed velocity which may be either ±v with equal probability, and a finite diffusivity. In one dimension, the interplay between convection and diffusion leads to a decay of the density which is proportional to t−3/4. At long times, the reactants organize into domains of rightand left-moving particles, with the typical distance between particles in a single domain growing as t3/4, and the distance between domains growing as t . The probability that an arbitrary particle reacts with its nth neighbour is found to decay as n−5/2 for same-velocity pairs and as n−7/4 for +− pairs. These kinetic and spatial exponents and their interrelations are obtained by scaling arguments. Our predictions are in excellent agreement with numerical simulations. Single-species diffusion-controlled annihilation, A + A → 0, exhibits classical mean-field kinetics when the spatial dimension d > 2, in which the concentration c(t) decays as t−1, and non-classical dimension-dependent kinetics for d 6 2 with a slower concentration decay, c(t) ∝ t−d/2 [1–7]. In one dimension, the geometric restriction to nearestneighbour interactions leads to relatively large departure from the mean-field kinetics, as well as a spatial organization of reactants. In this well-studied case, it is known that c(t) asymptotically decays as (Dt)−1/2, independent of the initial concentration. The complementary situation of single-species annihilation where the reactants move ballistically has recently begun to receive attention [8–12]. Perhaps the simplest example is the deterministic ± annihilation process, where each particle moves at a constant velocity which may be either +v or −v [8, 9]. When the densities of the +v and −v particles are equal, c(t) decays as (c0/vt). In this letter, we consider single species annihilation when the particle transport is a superposition of convection and diffusion—we term this system the stochastic ± annihilation process (figure 1). Although the concentration decays as t−1/2 when only one of the transport mechanisms—either convection or diffusion—is operative, the combined transport process leads to a faster concentration decay of t−3/4 [10]. Our goal is to understand this unusual decay law and its attendant consequences on the spatial distribution of reactants. While there has been fragmentary mention of some aspects of this system [7, 10], here we give primarily new results and a self-contained account of the basic phenomena. To set the stage for our approaches and results in the stochastic ± annihilation process, it is first helpful to provide a simple derivation for the decay of c(t) in the deterministic 0305-4470/96/220561+08$19.50 c © 1996 IOP Publishing Ltd L561 L562 Letter to the Editor Figure 1. Spacetime evolution of particles in the stochastic ± annihilation process. ± process. Let us consider a system where particles are placed with concentration c0 in a box of size L, and denote by c(L, t) the time-dependent concentration. Initially, there are N = c0L particles, and the difference between the number of rightand left-moving particles is of the order of 1N = |N+ −N−| ∼ √ N . Eventually, all particles who belong to the minority-velocity species are annihilated and thus c(L, t = ∞) ∼ 1N/L ∼ (c0/L). We assume a scaling form for the concentration, c(L, t) ∼ (c0/L)f (z) with z = L/vt . According to the above argument, f (z) → constant in the z → 0 limit. Conversely, in the short time limit, z → ∞, the concentration cannot depend on the box size, so that f (z) must be proportional to z1/2. Thus we find