Coagulation reactions in low dimensions: Revisiting subdiffusive $A+A$ reactions in one dimension

We present a theory for the coagulation reaction $A+A\to A$ for particles moving subdiffusively in one dimension. Our theory is tested against numerical simulations of the concentration of $A$ particles as a function of time (“anomalous kinetics”) and of the interparticle distribution function as a function of interparticle distance and time. We find that the theory captures the correct behavior asymptotically and also at early times, and that it does so whether the particles are nearly diffusive or very subdiffusive. We find that, as in the normal diffusion problem, an interparticle gap responsible for the anomalous kinetics develops and grows with time. This corrects an earlier claim to the contrary on our part.

Figure 1

(Color online) Reactant concentration vs time on a lattice of ${10}^4$ sites with periodic boundary conditions. $c\left(0\right)=1$, $\alpha =0.5$. Symbols give results of numerical simulations with kill (open symbols) and no-kill (black symbols) protocols. Circles: Mittag-Leffler-based waiting time distribution. Squares: Pareto waiting time distribution. Solid lines: our theory, Eq. (40), with $K_{alpha,P}$ (upper line) and with $K_{\alpha ,ML}$ (lower line). Broken lines: approximations $c\left(t\right)\sim 1/S_P\left(t\right)$ (upper) and $c\left(t\right)\sim 1/S_{ML}\left(t\right)$ (lower). For a definition of symbols and for quantitative fits to the various lines see text. In addition, in all the figures of this paper the error bars are smaller than the symbol sizes.

Figure 2

(Color online) Reactant concentration vs time on a lattice of ${10}^4$ sites with periodic boundary conditions. $c\left(0\right)=0.1$, $\alpha =0.5$, showing transient behavior. Symbols give results of Numerical simulations with “kill” protocol for Pareto (squares) and Mittag-Leffler-based (circles) waiting time distribution. Solid lines: our theory, Eq. (40), with $K_{\alpha ,P}$ (upper line) and $K_{\alpha ,ML}$ (lower line). Broken lines: asymptotic approximation for $c\left(t\right)$ as given in Eq. (31) for both cases.

Figure 3

(Color online) Asymptotic interparticle distribution function and empty interval function vs the scaled variable $z=c\left(t\right)x$ for the classic reaction-diffusion problem. The simulation results (symbols) are obtained with an exponential waiting time distribution on a lattice of $40000$ sites at time $t=6000$. The solid curves from the traditional exact theories are given in Eqs. (47, 48). The upper panel shows the interparticle distribution function and the inset shows it on a logarithmic scale that exhibits the simulation scatter at extremely low densities. The lower panel shows the empty interval function.

Figure 4

(Color online) Asymptotic interparticle distribution function and empty interval function as a function of the scaled variable $z=c\left(t\right)x$ for $\alpha =0.7$. The simulation results (symbols) are obtained with a Pareto waiting time distribution and a kill rule on a lattice of $40000$ sites at time $t={10}^6$. The solid theoretical curves are obtained from Eqs. (33, 30). The upper panel shows the interparticle distribution function and the inset shows it on a logarithmic scale that exhibits the simulation scatter at extremely low densities. The lower panel shows the empty interval function.

Figure 5

(Color online) Asymptotic interparticle distribution function and empty interval function as a function of the scaled variable $z=c\left(t\right)x$ for $\alpha =0.2$. The simulation results (symbols) are obtained with a Pareto waiting time distribution and a kill rule on a lattice of $40000$ sites at time $t={10}^{19}$. The solid theoretical curves are obtained from Eqs. (33, 30). The upper panel shows the interparticle distribution function and the inset shows it on a logarithmic scale that exhibits the simulation scatter at extremely low densities. The lower panel shows the empty interval function.

Figure 6

(Color online) Interparticle distribution function $p\left(z\right)$. Symbols: simulation results with a Pareto waiting time distribution. Squares (red in color): “kill” rule. Circles (blue in color): “no-kill” rule. Lattice size $L=10000,c\left(0\right)=1,c\left(t_{final}\right)=0.0025$. Upper panel: $\alpha =0.2$. Middle panel: $\alpha =0.5$. Lower panel: $\alpha =0.8$. The solid curves are calculated from Eq. (33).

Figure 7

(Color online) Interparticle distribution function for $\alpha =0.5$. Symbols: simulation results with a Mittag-Leffler waiting time distribution with $C\left(0\right)=0.1$. Upper panel: $t={10}^2$. Middle panel: $t={10}^5$. Lower panel: $t={10}^8$. The dashed curves are asymptotic, calculated from Eq. (33). The solid curves are obtained from Eq. (44).