Effects of intrinsic noise on a cubic autocatalytic reaction diffusion system

Abstract

Starting from our recent chemical master equation derivation of the model of an autocatalytic reaction-diffusion chemical system with reactions $U+2V {\stackrel {λ_0}{\rightarrow}}~ 3 V;$ and $V {\stackrel μ{\rightarrow}}~P$, $U {\stackrel ν{\rightarrow}}~ Q$, we determine the effects of intrinsic noise on the momentum-space behavior of its kinetic parameters and chemical concentrations. We demonstrate that the intrinsic noise induces $n \rightarrow n$ molecular interaction processes with $n \geq 4$, where $n$ is the number of molecules participating of type $U$ or $V$. The momentum dependences of the reaction rates are driven by the fact that the autocatalytic reaction (inelastic scattering) is renormalized through the existence of an arbitrary number of intermediate elastic scatterings, which can also be interpreted as the creation and subsequent decay of a three body composite state $σ= ϕ_u ϕ_v^2$, where $ϕ_i$ corresponds to the fields representing the densities of $U$ and $V$. Finally, we discuss the difference between representing $σ$ as a composite or an elementary particle (molecule) with its own kinetic parameters. In one dimension we find that while they show markedly different behavior in the short spatio-temporal scale, high momentum (UV) limit, they are formally equivalent in the large spatio-temporal scale, low momentum (IR) regime. On the other hand in two dimensions and greater, due to the effects of fluctuations, there is no way to experimentally distinguish between a fundamental and composite $σ$. Thus in this regime $σ$ behave as an entity unto itself suggesting that it can be effectively treated as an independent chemical species.