The kinetics of the q species pair annihilation reaction (A i + A j → ∅ for 1 ≤ i < j ≤ q) in d dimensions is studied by means of analytical considerations and Monte Carlo simulations. In the long-time regime the total particle density decays as ρ(t) ∼ t −α . For d = 1 the system segregates into single species domains, yielding a different value of α for each q; for a simplified version of the model in one dimension we derive α(q) = (q − 1)/(2q). Within mean-field theory, applicable in d ≥ 2, segregation occurs only for q < 1 + (4/d). The only physical realisation of this scenario is the twospecies process (q = 2) in d = 2 and d = 3, governed by an extra local conservation law. For d ≥ 2 and q ≥ 1 + (4/d) the system remains disordered and its density is shown to decay universally with the mean-field power law (α = 1) that also characterises the single-species annihilation process A + A → ∅.

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.PACS 05.40.-a, 82.20.-w