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.PACS 05.40.-a, 82.20.-w

Phase Transitions and Scaling in Systems Far from Equilibrium

“…is preserved by the reactions A + B → ∅ in one dimension; hence the distinction between A and B particles becomes meaningless, and their densities indeed satisfy the single-species t −1/2 pair annihilation power law. One may in fact fully analyze the q-species pair annihilations Ai + Aj → ∅, 1 ≤ i < j ≤ q, with equal initial densities ai(0) as well as uniform diffusion and reaction rates (141,142,143). Indeed, for more than two species (q > 2), there exists no conservation law in the stochastic kinetics, and species segregation results for d < ds(q) = 4/(q − 1).…”

Strongly anisotropic nonequilibrium phase transition in Ising models with friction

“…In this limit the class of models show mean field-like critical behavior. Subsequent investigations have been done in a variety of context, in particular for driven Potts models [7] and for rotating Ising chains of finite length [8].…”

Prebiotic Aggregates (Tissues) Emerging from Reaction–Diffusion: Formation Time, Configuration Entropy and Optimal Spatial Dimension

“…A boundary, a spatial (hyper-) cube, is assumed for mathematical convenience. As usual in reaction–diffusion equations [ 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 ], the stability of the background solution is related to the perturbation functions and . Algebraic equations for the stability parameter are then obtained from Equations (1) and (2): …”

“…Here, the wave vector (integers) defines a typical wavelength . The instability [ 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 ] of the homogenous solution, correlated with proto-tissue creation, is analyzed in the following sections.…”

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