Renormalization Group Study of the A+B->0 Diffusion-Limited Reaction

Abstract

The $A + B\to 0$ diffusion-limited reaction, with equal initial densities $a(0) = b(0) = n_0$, is studied by means of a field-theoretic renormalization group formulation of the problem. For dimension $d > 2$ an effective theory is derived, from which the density and correlation functions can be calculated. We find the density decays in time as $a,b \sim C\sqrt{\D}(Dt)^{-d/4}$ for $d < 4$, with $\D = n_0-C^\prime n_0^{d/2} + \dots$, where $C$ is a universal constant, and $C^\prime$ is non-universal. The calculation is extended to the case of unequal diffusion constants $D_A \neq D_B$, resulting in a new amplitude but the same exponent. For $d \le 2$ a controlled calculation is not possible, but a heuristic argument is presented that the results above give at least the leading term in an $ε= 2-d$ expansion. Finally, we address reaction zones formed in the steady-state by opposing currents of $A$ and $B$ particles, and derive scaling properties.