Steady-state diffusion-controlled A+A→0 reaction in Euclidean and fractal dimensions: Rate laws and particle self-ordering

For diffusion-limited reactions of the type A+A{→}0, we present a theory for the pair correlation function and the macroscopic reaction law for the Euclidean dimensions d=1, 2, 3 and for self-similar fractal structures with spectral dimension such that 1{≤}dₛ{≤}2. In any dimension we define a length {Λ}, which is the typical size of the depletion zone around each particle. {Λ} is expressed as a function of a fundamental size {ξ}= {}{ρ}D/R , where {ρ} is the steady-state density, R the external source rate, and D the microscopic diffusion constant. For dₛ{≤}2, we obtain the ``fractal'' reaction order X=1+2/dₛ and a mesoscopic, density-dependent depletion zone around each particle. We give an interpretation of the particle self-ordering and anomalous rate laws in terms of the random-walk topological properties.