Effects of intrinsic noise on a cubic autocatalytic reaction-diffusion system

Starting from our recent chemical master equation derivation of the model of an autocatalytic reaction-diffusion chemical system with reactions $U+2V\stackrel{{\lambda }_0}{\to }3V$ and $V\stackrel{\mu }{\to }P$, $U\stackrel{\nu }{\to }Q$, 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 $n\to n$ molecular interaction processes with $n\ge 4$, where $n$ is the number of participating molecules 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 $\sigma ={\varphi }_u{\varphi }_v^2$, where ${\varphi }_i$ corresponds to the fields representing the densities of $U$ and $V$. Finally, we discuss the difference between representing $\sigma$ 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 spatiotemporal scale, high-momentum (UV) limit, they are formally equivalent in the large spatiotemporal 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 $\sigma$. Thus, in this regime, $\sigma$ behaves as an entity unto itself, suggesting that it can be effectively treated as an independent chemical species.

Figure 1

Many-body interpretation of the reaction $U+2V\to 3V$. (a) Inelastic scattering where one molecule of $U$ and two molecules of $V$ are destroyed at an interaction vertex (shaded) to create three molecules of $V$. (b) The elastic scattering where $U+2V\to U+2V$, which must be present in order for particle number and probability conservation. Time flows from right to left.

Figure 2

The diagrammatic expansion for the vertex function ${\Gamma }^{(3,3)}$ for inelastic scattering. The same set of primitive divergences contributes to that for elastic scattering, which is the same except for the substitution of a single outgoing leg from ${\varphi }_v$ to ${\varphi }_u$. Consequently, both processes are renormalized through a single renormalized coupling constant.

Figure 3

The $\beta$ function (15) as a function of the dimensionless coupling $g_R\left({\tilde{s}}_0\right)$. Note that this refers to IR and not UV flow. There are two fixed points in the regions $d<1$ and $2<d<3$, whereas there is a single fixed point when $1<d<2$ and $d>3$.

Figure 4

The vertex function ${\Gamma }_{\mathrm{i}}^{(3,3)}$ shown in Fig. 2 reinterpreted as the production and decay of the composite field $\sigma$. The composite field $\sigma$ is formed out of the combination $\lambda {\varphi }_u{\varphi }_v^2$ and leads to the two decay channels shown in (a) and (b).

Figure 5

Fluctuation induced $n\to n$ scattering for (a) $n=4,2U+2V\to U+3V$ and (b) $n=5,3U+2V\to 2U+3V$. The shaded circle represents the renormalized original $3\to 3$ scattering from Fig. 2. These processes proceed through the single loop consisting of two ${\varphi }_v$'s, which can be thought of as a composite field correlation function for ${\psi }_1={\varphi }_v^2$. We can have related scattering processes through the composite field correlation function for ${\psi }_2={\varphi }_u{\varphi }_v$, as shown in (c) for the reaction $U+3V\to 4V$ and, indeed, a combination of the two composite field correlation functions, as shown in (d) for the reaction $2U+3V\to U+4V$.