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We investigate the density decay in the pair-annihilation process A+A→∅ in the case when the particles perform anomalous diffusion on a cubic lattice. The anomalous diffusion is realized via Lévy flights, which are characterized by long-range jumps and lead to superdiffusive behavior. As a consequence, the critical dimension depends continuously on the control parameter of the Lévy flight distribution. This instance is used to study the system close to the critical dimension by means of the nonperturbative renormalization group theory. Close to the critical dimension, the assumption of well-stirred reactants is violated by anticorrelations between the particles, and the law of mass action breaks down. The breakdown of the law of mass action is known to be caused by long-range fluctuations. We identify three interrelated consequences of these fluctuations. First, despite being a nonuniversal quantity and thus depending on the microscopic details, the renormalized reaction rate λ(0) can be approximated by a universal law close to the critical dimension. The emergence of universality relies on the fact that long-range fluctuations suppress the influence of the underlying microscopic details. Second, as criticality is approached, the macroscopic reaction rate decreases such that the law of mass action loses its significance. And third, additional nonanalytic power law corrections complement the analytic law of mass action term. An increasing number of those corrections accumulate and give an essential contribution as the critical dimension is approached. We test our findings for two implementations of Lévy flights that differ in the way they cross over to the normal diffusion in the limit σ→2.

We study the problem of diffusing particles which coalesce upon contact. With the aid of a nonperturbative renormalization group, we first analyze the dynamics emerging below the critical dimension two, where strong fluctuations imply anomalously slow decay. Above two dimensions, the long-time, low-density behavior is known to conform with the law of mass action. For this case, we establish an exact mapping between the physics at the microscopic scale (lattice structure, particle shape and size) and the macroscopic decay rate in the law of mass action. In addition, we identify a term violating this classical law. It originates in long-range and many-particle fluctuations and is a simple, universal function of the macroscopic decay rate.

Diffusion-limited reactions are studied in detail on the classical coalescing process. We demonstrate how, with the aid of a recent renormalization group approach, fluctuations can be integrated systematically. We thereby obtain an exact relation between the microscopic physics (lattice structure and particle shape and size) and the macroscopic decay rate in the law of mass action. Moreover, we find a strong violation of the law of mass action. The corresponding term in the kinetic equations originates in long-wavelength fluctuations and is a universal function of the macroscopic decay rate.

Cyclic (rock-paper-scissors-type) population models serve to mimic complex species interactions. Focusing on a paradigmatic three-species model with mutations in one dimension, we observe an interplay between equilibrium and nonequilibrium processes in the stationary state. We exploit these insights to obtain asymptotically exact descriptions of the emerging reactive steady state in the regimes of high and low mutation rates. The results are compared to stochastic lattice simulations. Our methods and findings are potentially relevant for the spatiotemporal evolution of other nonequilibrium stochastic processes.