Prof John Cardy FRS

Abstract:

We study theoretically and numerically the steady state diffusion controlled reaction [Formula Presented], where currents [Formula Presented] of [Formula Presented] and [Formula Presented] particles are applied at opposite boundaries. For a reaction rate [Formula Presented], and equal diffusion constants [Formula Presented], we find that when [Formula Presented] the reaction front is well described by mean-field theory. However, for [Formula Presented], the front acquires a Gaussian profile—a result of noise induced wandering of the reaction front center. We make a theoretical prediction for this profile which is in good agreement with simulation. Finally, we investigate the intrinsic (nonwandering) front width and find results consistent with scaling and field theoretic predictions. © 1996 The American Physical Society.

Abstract:

A systematic theory for the diffusion-limited reaction processes A + A → 0 and A → (m + 1) A is developed. Fluctuations are taken into account via the field-theoretic dynamical renormalization group. For even m, the mean field rate equation, which predicts only an active phase, remains qualitatively correct near dc = 2 dimensions; but below d'c≈4/3 a nontrivial transition to an inactive phase governed by power law behavior appears. For odd m, there is a dynamic phase transition for any d ≤ 2 which is described by the directed percolation universality class. © 1996 The American Physical Society.

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.