Diffusion - Limited Aggregation Processes with 3 - Particle Elementary Reactions

A diffusion-limited aggregation process, in which clusters coalesce by means of 3-particle reaction, A + A + A → A, is investigated. In one dimension we give a heuristic argument that predicts logarithmic corrections to the mean-field asymptotic behavior for the concentration of clusters of mass m at time t, C m (t) ∼ m −1/2 (log(t)/t) 3/4 , for 1 ≪ m ≪ t/ log(t). The total concentration of clusters, C(t), decays as C(t) ∼ log(t)/t at t → ∞. We also investigate the problem with a localized steady source of monomers and find that the steady-state concentration C(r) scales as r −1 (log(r)) 1/2 , r −1 , and r −1 (log(r)) −1/2 , respectively , for the spatial dimension d equal to 1, 2, and 3. The total number of clusters, N (t), grows with time as (log(t)) 3/2 , t 1/2 , and t(log(t)) −1/2 for d = 1, 2, and 3. Furthermore, in three dimensions we obtain an asymptotic solution for the steady state cluster-mass distribution: C m (r) ∼ r −1 (log(r)) −1 Φ(z), with the scaling function Φ(z) = z −1/2 exp(−z) and the scaling variable z ∼ m/ log(r).