Reaction-controlled diffusion

The dynamics of a coupled two-component nonequilibrium system is examined by means of continuum field theory representing the corresponding master equation. Particles of species A may perform hopping processes only when particles of different type B are present in their environment. Species B is subject to diffusion-limited reactions. If the density of B particles attains a finite asymptotic value (active state), the A species displays normal diffusion. On the other hand, if the B density decays algebraically ∝ t^−α at long times (inactive state), the effective attractive A-B interaction is weakened. The combination of B decay and activated A hopping processes gives rise to anomalous diffusion, with mean-square displacement <x_A(t)² ∝ t^1−α for α < 1. Such algebraic subdiffusive behavior ensues for n-th order B annihilation reactions (nB → ∅) with n ≥ 3, and n = 2 for d < 2. The mean-square displacement of the A particles grows only logarithmically with time in the case of B pair annihilation (n = 2) and d ≥ 2 dimensions. For radioactive B decay (n = 1), the A particles remain localized. If the A particles may hop spontaneously as well, or if additional random forces are present, the A-B coupling becomes irrelevant, and conventional diffusion is recovered in the long-time limit.