Kinetics of diffusion-limited bimolecular reactions in low-dimensional media.

Abstract

We investigate three cases of bimolecular reaction kinetics in the diffusion limited regime when the medium is not necessarily a three dimensional euclidean space. This work is a theoretical approach that connects the fluctuations of reactant distribution at steady state, to the fundamental properties of a random walk. We use the fractal model as a topological extension of euclidean spaces. We show that for spectral dimensions below the critical dimension d$\sb{\rm s}$ = 2, a self-organization of reactants takes place in the system. The three examples of biomolecular reaction are the homomolecular annihilation A+A$\to$0, the heteromolecular annihilation A+B$\to$0 and the trapping problem A+T$\to$T where T is a fixed trap. In the A+A$\to$0 case we observe a mesoscopic self-organization characterized by an averaged depletion zone around each particle and an anomalous order of reaction X = 1 + 2/d$\sb{\rm s}$, in the low density limit. In the A+B$\to$0 case we observe a segregation that may be microscopic, mesoscopic or macroscopic according to the nature of the source term. We separate two types of sources, the ones with a strict conservation and the ones with a statistical conservation in the number of As and Bs in the system. In the strict conservation case we have a classical order of reaction X = 2 and segregation to a size determined by the nature of the source. In this case we study sources with random and correlated A-B separations. In the statistical conservation case, we have a non-reactive steady state with a saturation of the system in one of the reactants. If a vertical annihilation mechanism is present, we find segregation to a scale fixed by the external rate of particles and an effective order of reaction X = 4/d$\sb{\rm s}$. If a first order symmetric decay mechanism A$\to$0 and B$\to$0 is present, we find a mesoscopic segregation defined by the decay rate constant and a classical order of reaction X = 2. For the trapping problem, we find for d$\sb{\rm s} <$ 2, an anomalous partial order of reaction relative to the trap concentration X = 2/d$\sb{\rm s}$. The theoretical predictions are tested via Monte-Carlo simulations on lattices.