Distribution in flowing reaction-diffusion systems

A power-law distribution is found in the density profile of reacting systems $A+B\to C+D$ and $2A\to 2C$ under a flow in two and three dimensions. Different densities of reactants $A$ and $B$ are fixed at both ends. For the reaction $A+B$, the concentration of reactants asymptotically decay in space as $x^{-1/2}$ and $x^{-3/4}$ in two dimensions and three dimensions, respectively. For $2A$, it decays as $\text{log}\left(x\right)/x$ in two dimensions. The decay of $A+B$ is explained considering the effect of segregation of reactants in the isotropic case. The decay for $2A$ is explained by the marginal behavior of two-dimensional diffusion. A logarithmic divergence of the diffusion constant with system size is found in two dimensions.

Figure 1

(Color online) Log-log plot of spatial distributions of reactants of the reaction $A+B\to C+D$ in two and three dimensions. The parameters are fixed as ${\overline{\rho }}_L=0.155$ and ${\overline{\rho }}_R=0.111$, $L_x=100$, and $L_y=L_z=10$. The lines $x^{-1/2}$ and $x^{-3/4}$ are also shown. In the inset, we show a zoom of the profile in which we can see the decrease in reactants and an increase in products.

Figure 2

Snapshots in our geometry. The parameters of the simulation are $L_x=800$ and $L_y=100$. Black and white circles denote $A$ and $B$, respectively. The left end corresponds to the high-density side. Radii of the reactants are shown enlarged to make the presentation clearer.

Figure 3

(Color online) Particle-particle correlation functions $g_{AA}^x\left(r\right)$. In the direction of the arrow, $g_{AA}^x$ for $x=5,6,\dots ,13$ are shown. The right end $r=5$ is equal to $L_y/2$. In the inset, we show the values at $r=L_y/2$ as a function of $x$. They are fitted by $f\left(x\right)=1+\frac{C_1}{\sqrt{x}}\text{exp}\left(-C_2/x\right)$ with $C_1=11.4$ and $C_2=31.6$.

Figure 4

(Color online) Distribution of reactant $A$ in the reaction $A+B=C+D+Q$ with energy release $Q$ in two dimensions for different $Q$. The densities asymptotically approach $x^{-1/2}$.

Figure 5

(Color online) Distribution of particles in $2A\to 2C$. The line $x^{-1}$ is shown as the mean-field solution. Reactant $A$ of reactions $2A$ appears to decay slightly slower and approach $\text{log}\left(x\right)/x$.

Figure 6

(Color online) Dependence of the diffusion constant on system size.