Symmetry and species segregation in diffusion-limited pair annihilation

TL;DR

This paper investigates how symmetry and reaction rates influence species segregation and density decay in diffusion-limited annihilation systems, identifying conditions for segregation and deriving decay laws through analytical and simulation methods.

Contribution

It establishes the conditions under which symmetry-breaking and segregation occur in multi-species annihilation, linking them to the eigenvalues of the reaction rate matrix.

Findings

Existence of a critical segregation dimension d_{seg} depends on the eigenvalues of k.

Density decay slows down below d_{seg} to a power law determined by eigenvalue ratios.

Segregation can occur without a conservation law, contrary to previous assumptions.

Abstract

We consider a system of q diffusing particle species A_1,A_2,...,A_q that are all equivalent under a symmetry operation. Pairs of particles may annihilate according to A_i + A_j -> 0 with reaction rates k_{ij} that respect the symmetry, 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 conditions on the matrix k under which there exists a critical segregation dimension d_{seg} 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 n(t) ~ t^{-d/d_{seg}} for 2 < d < d_{seg}. We show that when d_{seg} 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.