Symmetry and species segregation in diffusion-limited pair annihilation

Abstract

We consider a system of q diffusing particle species A1, A2, ..., Aq that are all equivalent under a symmetry operation. Pairs of particles may annihilate according to Ai + Aj → 0 with reaction rates kij that respect the symmetry, and without self-annihilation (kii = 0). In spatial dimensions d > 2 mean-field theory predicts that the total particle density decays as ρ(t) ∼ t-1 , provided the system remains spatially uniform. We determine the conditions on the matrix k under which there exists a critical segregation dimension dseg below which this uniformity condition is violated; the symmetry between the species is then locally broken. We argue that in those cases the density decay slows down to ρ(t) ∼ t−d/dseg for 2 < d < dseg. We show that when dseg exists, its value can be expressed in terms of the ratio of the smallest to the largest eigenvalue of k. The existence of a conservation law (as in the special two-species annihilation A + B → 0), although sufficient for segregation, is shown not to be a necessary condition for this phenomenon to occur. We work out specific examples and present Monte Carlo simulations compatible with our analytical results.