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We consider a system of q diffusing particle species A_1,A_2,...,A_q that areall equivalent under a symmetry operation. Pairs of particles may annihilateaccording to A_i + A_j -> 0 with reaction rates k_{ij} that respect thesymmetry, and without self-annihilation (k_{ii} = 0). In spatial dimensions d >2 mean-field theory predicts that the total particle density decays as n(t) ~1/t, provided the system remains spatially uniform. We determine the conditionson the matrix k under which there exists a critical segregation dimensiond_{seg} below which this uniformity condition is violated; the symmetry betweenthe species is then locally broken. We argue that in those cases the densitydecay slows down to n(t) ~ t^{-d/d_{seg}} for 2 < d < d_{seg}. We show thatwhen d_{seg} exists, its value can be expressed in terms of the ratio of thesmallest to the largest eigenvalue of k. The existence of a conservation law(as in the special two-species annihilation A + B -> 0), although sufficientfor segregation, is shown not to be a necessary condition for this phenomenonto occur. We work out specific examples and present Monte Carlo simulationscompatible with our analytical results.Comment: latex, 19 pages, 3 eps figures include

Symmetry and species segregation in diffusion-limited pair annihilation