Exponential temporal asymptotics of the $A+B\to 0$ reaction-diffusion process with initially separated reactants

We study theoretically and numerically the irreversible $A+B\to 0$ reaction-diffusion process of initially separated reactants occupying the regions of lengths $L_A$, $L_B$ comparable with the diffusion length ($L_A,L_B\sim \sqrt{Dt}$, here $D$ is the diffusion coefficient of the reactants). It is shown that the process can be divided into two stages in time. For $t⪡L^2∕D$ the front characteristics are described by the well-known power-law dependencies on time, whereas for $t>L^2∕D$ these are well-approximated by exponential laws. The reaction-diffusion process of about 0.5 of initial quantities of reactants is described by the obtained exponential laws. Our theoretical predictions show good agreement with numerical simulations.

Figure 1

Numerical simulation of the reaction-diffusion process for the case of equal diffusion coefficients, $D=D_A=D_B=0.5$ (the probabilities of unit jumps to the left and to the right are 0.5) and $L_A=L_B=L=50$: (a) the distribution of the particle density at $t=1100$ [solid lines are given by Eq. (8)]; (b) the temporal dependencies of the total and (c) maximum reaction rates; and (d) the reaction zone width. Inset in (b) presents the temporal dependence of the total reaction rate at $t<\sim t_b$. Reaction probability is 0.0001; the initial particle concentrations are 100 000. Note that for the chosen parameters $D$ and $L$ the coefficient $a$ in Eqs. (9, 10, 11) is 0.000 493 and analytically predicted the exponent indexes of $R\left(t\right)$, $R_f\left(t\right)$, and $w\left(t\right)$ are $-0.000493$, $-0.000657$, and $+0.000164$, respectively.

Figure 2

Numerical simulation of the reaction-diffusion process for the case of immovable $B$ particles. The initial particle concentrations are $N_{B0}=100000$, $N_{A0}=50000$, and $D_B=0$, $D_A=0.5$, and initial lengths $L_A=L_B=20$. The distribution of the particle density at $t=2000$: (a) the temporal dependencies of the total (b) and maximum (c) reaction rates and of the reaction zone width (d). Inset in (b) presents the temporal dependence of the total reaction rate at $t\lesssim t_b$. Reaction probability is 0.000 01. In the asymptotic time regime the length occupied by the $A$ reactant is about 29 [see (a)]. The analytically estimated exponent index $a_A$ of $R\left(t\right)$ and $R_f\left(t\right)$ is 0.001 47 for the chosen parameters $D$ and $L_A$ and the front width is independent of time