Prof John Cardy FRS

Abstract:

We consider the properties of the diffusion-controlled reaction A+B to OE in the steady state, where fixed currents of A and B particles are maintained at opposite edges of the system. Using renormalization-group methods, we explicitly calculate the asymptotic forms of the reaction front and particle densities as expansions in (JD-1 mod x mod d+1) -1, where J are the (equal) applied currents, and D the (equal) diffusion constants. For the asymptotic densities of the minority species, we find, in addition to the expected exponential decay, fluctuation-induced power-law tails, which, for d<2, have a universal form A mod x mod - mu, where mu =5+O( epsilon ), and epsilon =2-d. A related expansion is derived for the reaction rate profile R, where we find the asymptotic power law R approximately B mod x mod - mu -2. For d>2, we find similar power laws with mu =d+3, but with non-universal coefficients. Logarithmic corrections occur in d=2. These results imply that, in the time-dependent case, with segregated initial conditions, the moments integral mod x mod qR(x,t) dx fail to satisfy simple scaling for q> mu +1. Finally, it is shown that the fluctuation-induced wandering of the position of the reaction front centre may be neglected for large enough systems.

Abstract:

The A+B→{circled division slash} diffusion-limited reaction, with equal initial densities a(0)=b(0)=n0, 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 call be calculated. We find the density decays in time as {Mathematical expression} for d<4, with Δ=n0-C′n0d/2+..., where C is a universal constant and C′ is nonuniversal. The calculation is extended to the case of unequal diffusion constants DA≠DB, resulting in a new amplitude but the same exponent. For d≤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. © 1995 Plenum Publishing Corporation.