Kinetic regimes and cross-over times in many-particle reacting systems

– We study kinetics of single-species reactions (“A + A → ∅”) for general local reactivity Q and dynamical exponent z (rms displacement xt ∼ t.) For small molecules z = 2, whilst z = 4, 8 for certain polymer systems. For dimensions d above the critical value dc = z, kinetics are always mean field (MF). Below dc, the density nt initially follows MF decay, n0 − nt ∼ n 2 0Qt. A 2-body diffusion-controlled regime follows for strongly reactive systems (Q > Q ∼ n 0 ) with n0 − nt ≈ n 2 0x d t . For Q < Q , MF kinetics persist, with nt ∼ 1/Qt. In all cases nt ≈ 1/x d t at the longest times. Our analysis avoids decoupling approximations by instead postulating weak physically motivated bounds on correlation functions. The kinetics of reactions between diffusing particles are anomalous in low spatial dimensions [1, 2]. Mean-field (MF) theory, according to which the reaction rate is proportional to the product of reactant densities, is only valid for dimensions d above a critical value dc. In the most fundamental problem of single-species reactions into inert products, A+A→ ∅, for d < dc = 2 the density decays asymptotically as 1/t , independently of the magnitude of the local chemical reactivity Q and initial reactant density. That is, in low dimensions the 1/Qt decay predicted by MF theory is invalid. This is supported by numerical simulations [2-5] and renormalization group studies [6]. For the case of “infinitely” reactive particles for which Q = ∞ (probability of reaction per collision unity), rigorous bounds [7] on the asymptotic decay of density have been derived and in one dimension exact solutions exist [8-11]. Many other workers have employed approximate methods starting from the hierarchy of coupled dynamical equations for correlation functions of all orders which is then truncated by expressing higher-order correlation functions in terms of lower-order correlations [1, 12]. In this work we address the two major aspects of single-species reaction kinetics which remain poorly understood. 1) Although it is known that at the shortest timescales MF kinetics apply [13,14], the crossover from the short-time MF kinetics to the asymptotic 1/t behavior has not been established. Some groups on the basis of numerical simulations [15-17]